2.4 The Precise Definition of a Limit

Size: px
Start display at page:

Download "2.4 The Precise Definition of a Limit"

Transcription

1 2.4 The Precise Definition of a Limit Reminders/Remarks: x 4 < 3 means that the distance between x and 4 is less than 3. In other words, x lies strictly between 1 and 7. So, x a < δ means that the distance between x and a is less than δ, that is, x lies within δ of a. Note: The inequality 0 < x a < δ means the distance between x and a is less than δ, but x a. Similarly, if we write f(x) L < 2, this means that f(x) lies within 2 of L. Consider the function f(x) = 2x + 4. We know x 3 (2x + 4) = 10. We said that the it means we can make f(x) as close to 10 as we want by getting x closer and closer to 3. How close to 3 does x need to be so that f(x) lies within 1 of the it value 10, i.e. f(x) 10 < 1? How close to 3 does x need to be so that f(x) lies within 0.1 of the it value 10, i.e. f(x) 10 < 0.1? How close to 3 does x need to be so that f(x) lies within an arbitrary number ǫ of 10, i.e. f(x) 10 < ǫ? This number that we re finding is denoted as δ. 1

2 ǫ,δ Definition of a Limit: Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the it of f(x) as x approaches a is L, and we write f(x) = L x a if for every number ǫ > 0 there exists a number δ > 0 such that whenever 0 < x a < δ, then f(x) L < ǫ. To prove a it using the definition of the it, there are two steps: 1. Do some scratchwork to determine a value for δ. 2. Show that this δ works. Example: Prove using the definition of the it that x 3 (2x+4) = 10. 2

3 Example: Prove using the definition of the it that (3x+7) = 1. x 2 Consider the function f(x) = x 2. We know x 2 x 2 = 4. Question: Given ǫ = 1, find a number δ so that if x 2 < δ, then x 2 4 < 1. Question: Given ǫ = 0.1, find a number δ so that if x 2 < δ, then x 2 4 <

4 Consider the function f(x) = 1 1 x2. We know =. We said that this means we can make f(x) as x 0 x2 large as we want by getting x closer and closer to 0. Definition: Left f be a function defined on some open interval that contains the number a, except possibly at a itself. Then x a f(x) = if for every number M > 0 there exists a number δ > 0 such that whenever 0 < x a < δ, then f(x) > M. Find a number δ so that if 0 < x < δ, then 1 x 2 > 100. Find a number δ so that if 0 < x < δ, then 1 x 2 > 10,000. Find a number δ so that if 0 < x < δ, then 1 > M, where M is some arbitrary number. x2 4

5 2.5 Continuity In Section 2.3 we saw that the it as x approaches a can sometimes be found by evaluating the function at a. If this is the case, then the function is continuous. Definition: A function is continuous at a number a if f(x) = f(a) x a Otherwise, we say the function is discontinuous at a, or that there is a discontinuity at a. In order for a function to be continuous at a number a: (1) f(a) must be defined. So a function will NOT be continuous anywhere it is undefined. (2) x a f(x) must exist. (The left-handed and right-handed its must both equal the same value.) (3) x a f(x) = f(a) Examples of discontinuities: Holes, vertical asymptotes, and jumps. A hole in a graph is also referred to as a removable discontinuity because if we wanted to, we could just redefine the function at that point to make it continuous. Removable discontinuities occur where the it exists at a (left and right its are equal), but is not equal to f(a). A vertical asymptote is referred to as an infinite discontinuity. A jump in the graph is referred to as a jump discontinuity. Jumps occur where the its from the left and right exist, but are not equal. 5

6 A function is continuous from the left at a number a if = f(a) and continuous from the x a f(x) right if x a +f(x) = f(a). A function is continuous if and only if it is continuous from both the right and the left. Examples: Determine where the functions below are discontinuous. State the type of discontinuity and explain why mathematically using its. Is the function continuous from the left or right there? (1) f(x) = x2 25 x 5 (2) f(x) = (x+3)(x 2) (x 2) 3 (3) f(x) = { x 2 4 if x 1 x+1 if x > 1 Fact: All polynomials are continuous everywhere! Fact: A rational function is continuous wherever it is defined, i.e. where the denominator is not 0. 6

7 2x 1 if x < 4 (4) f(x) = 6 if x = 4 x 2 9 if x > 4 (5) f(x) = 3x+1 if x < 2 x 2 5 x 1 if 2 x 3 x 3 25 x 2 if x > 3 What values of a, b, and c would make the following function continuous everywhere? ax 2 +bx+1 if x 1 3x+2a+b if 1 < x < 2 f(x) = c if x = 2 x 2 ax b if x > 2 7

8 If f and g are continuous at a and c is any constant, then the functions f +g, f g, cf, fg, and f g (where g(a) 0) are all continuous functions. The Intermediate Value Theorem: Suppose f is continuous on the closed interval [a, b] and let N be any number strictly between f(a) and f(b). Then there exists a number c in (a,b) such that f(c) = N. Example: Show that the equation x 3 +2x+2 = 0 has a root (solution) on the interval (1,2). Example: If f(x) = x 4 x 3 +3x 2 +2, show that there is a number c so that f(c) = 3. 8

9 2.6 Limits at Infinity; Horizontal Asymptotes Up to this point, we have dealt with its as x approaches some number a. Now, we examine its as x approaches or. These are called its at infinity. Let f be a function defined on some interval (a, ). Then, f(x) = L x means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. Definition: If f(x) = L or f(x) = L, then f(x) has a horizontal asymptote at y = L. x x 1 x x = 2 x x = 3 = x ± x 2 x (x3 x 2 ) x (x3 x 2 ) Evaluate the following its: 2x 3 +x 2 1 x 5x 3 7x+2 2x 3 +x 2 1 x 5x 3 7x+2 9

10 4x 2 2x+3 x 5 3x 4x 2 2x+3 x 5 3x x x 2 (x 8) (x 2 +1)(x 2 3) x x 2 (x 8) (x 2 +1)(x 2 3) x 9x x+2 x 9x x+2 10

11 x 2 x x x x 2 +5x+x x Find all horizontal asymptotes of the function f(x) = 5 4x3 25x

2.2 The Limit of a Function

2.2 The Limit of a Function 2.2 The Limit of a Function Introductory Example: Consider the function f(x) = x is near 0. x f(x) x f(x) 1 3.7320508 1 4.236068 0.5 3.8708287 0.5 4.1213203 0.1 3.9748418 0.1 4.0248457 0.05 3.9874607 0.05

More information

1.5 Inverse Trigonometric Functions

1.5 Inverse Trigonometric Functions 1.5 Inverse Trigonometric Functions Remember that only one-to-one functions have inverses. So, in order to find the inverse functions for sine, cosine, and tangent, we must restrict their domains to intervals

More information

MATH 151 Engineering Mathematics I

MATH 151 Engineering Mathematics I MATH 151 Engineering Mathematics I Spring 2018, WEEK 3 JoungDong Kim Week 3 Section 2.5, 2.6, 2.7, Continuity, Limits at Infinity; Horizontal Asymptotes, Derivatives and Rates of Change. Section 2.5 Continuity

More information

MATH 151 Engineering Mathematics I

MATH 151 Engineering Mathematics I MATH 151 Engineering Mathematics I Fall 2018, WEEK 3 JoungDong Kim Week 3 Section 2.3, 2.5, 2.6, Calculating Limits Using the Limit Laws, Continuity, Limits at Infinity; Horizontal Asymptotes. Section

More information

The function graphed below is continuous everywhere. The function graphed below is NOT continuous everywhere, it is discontinuous at x 2 and

The function graphed below is continuous everywhere. The function graphed below is NOT continuous everywhere, it is discontinuous at x 2 and Section 1.4 Continuity A function is a continuous at a point if its graph has no gaps, holes, breaks or jumps at that point. If a function is not continuous at a point, then we say it is discontinuous

More information

x y More precisely, this equation means that given any ε > 0, there exists some δ > 0 such that

x y More precisely, this equation means that given any ε > 0, there exists some δ > 0 such that Chapter 2 Limits and continuity 21 The definition of a it Definition 21 (ε-δ definition) Let f be a function and y R a fixed number Take x to be a point which approaches y without being equal to y If there

More information

Chapter 2: Functions, Limits and Continuity

Chapter 2: Functions, Limits and Continuity Chapter 2: Functions, Limits and Continuity Functions Limits Continuity Chapter 2: Functions, Limits and Continuity 1 Functions Functions are the major tools for describing the real world in mathematical

More information

CH 2: Limits and Derivatives

CH 2: Limits and Derivatives 2 The tangent and velocity problems CH 2: Limits and Derivatives the tangent line to a curve at a point P, is the line that has the same slope as the curve at that point P, ie the slope of the tangent

More information

Chapter 2. Limits and Continuity. 2.1 Rates of change and Tangents to Curves. The average Rate of change of y = f(x) with respect to x over the

Chapter 2. Limits and Continuity. 2.1 Rates of change and Tangents to Curves. The average Rate of change of y = f(x) with respect to x over the Chapter 2 Limits and Continuity 2.1 Rates of change and Tangents to Curves Definition 2.1.1 : interval [x 1, x 2 ] is The average Rate of change of y = f(x) with respect to x over the y x = f(x 2) f(x

More information

6.1 Polynomial Functions

6.1 Polynomial Functions 6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and

More information

Continuity. To handle complicated functions, particularly those for which we have a reasonable formula or formulas, we need a more precise definition.

Continuity. To handle complicated functions, particularly those for which we have a reasonable formula or formulas, we need a more precise definition. Continuity Intuitively, a function is continuous if its graph can be traced on paper in one motion without lifting the pencil from the paper. Thus the graph has no tears or holes. To handle complicated

More information

DRAFT - Math 101 Lecture Note - Dr. Said Algarni

DRAFT - Math 101 Lecture Note - Dr. Said Algarni 2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that

More information

Induction, sequences, limits and continuity

Induction, sequences, limits and continuity Induction, sequences, limits and continuity Material covered: eclass notes on induction, Chapter 11, Section 1 and Chapter 2, Sections 2.2-2.5 Induction Principle of mathematical induction: Let P(n) be

More information

Lecture 3 (Limits and Derivatives)

Lecture 3 (Limits and Derivatives) Lecture 3 (Limits and Derivatives) Continuity In the previous lecture we saw that very often the limit of a function as is just. When this is the case we say that is continuous at a. Definition: A function

More information

Review: Limits of Functions - 10/7/16

Review: Limits of Functions - 10/7/16 Review: Limits of Functions - 10/7/16 1 Right and Left Hand Limits Definition 1.0.1 We write lim a f() = L to mean that the function f() approaches L as approaches a from the left. We call this the left

More information

1.3 Limits and Continuity

1.3 Limits and Continuity .3 Limits and Continuity.3. Limits Problem 8. What will happen to the functional values of as x gets closer and closer to 2? f(x) = Solution. We can evaluate f(x) using x values nearer and nearer to 2

More information

Limits and Continuity

Limits and Continuity Limits and Continuity MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics Fall 2018 Objectives After this lesson we will be able to: Determine the left-hand and right-hand limits

More information

Limits and Continuity

Limits and Continuity Limits and Continuity Philippe B. Laval Kennesaw State University January 2, 2005 Contents Abstract Notes and practice problems on its and continuity. Limits 2. Introduction... 2.2 Theory:... 2.2. GraphicalMethod...

More information

Continuity. MATH 161 Calculus I. J. Robert Buchanan. Fall Department of Mathematics

Continuity. MATH 161 Calculus I. J. Robert Buchanan. Fall Department of Mathematics Continuity MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Fall 2017 Intuitive Idea A process or an item can be described as continuous if it exists without interruption. The mathematical

More information

MATH 151 Engineering Mathematics I

MATH 151 Engineering Mathematics I MATH 151 Engineering Mathematics I Fall, 2016, WEEK 4 JoungDong Kim Week4 Section 2.6, 2.7, 3.1 Limits at infinity, Velocity, Differentiation Section 2.6 Limits at Infinity; Horizontal Asymptotes Definition.

More information

Calculus I Exam 1 Review Fall 2016

Calculus I Exam 1 Review Fall 2016 Problem 1: Decide whether the following statements are true or false: (a) If f, g are differentiable, then d d x (f g) = f g. (b) If a function is continuous, then it is differentiable. (c) If a function

More information

Intermediate Value Theorem

Intermediate Value Theorem Stewart Section 2.5 Continuity p. 1/ Intermediate Value Theorem The intermediate value theorem states that, if a function f is continuous on a closed interval [a,b] (that is, an interval that includes

More information

Homework for Section 1.4, Continuity and One sided Limits. Study 1.4, # 1 21, 27, 31, 37 41, 45 53, 61, 69, 87, 91, 93. Class Notes: Prof. G.

Homework for Section 1.4, Continuity and One sided Limits. Study 1.4, # 1 21, 27, 31, 37 41, 45 53, 61, 69, 87, 91, 93. Class Notes: Prof. G. GOAL: 1. Understand definition of continuity at a point. 2. Evaluate functions for continuity at a point, and on open and closed intervals 3. Understand the Intermediate Value Theorum (IVT) Homework for

More information

Continuity, Intermediate Value Theorem (2.4)

Continuity, Intermediate Value Theorem (2.4) Continuity, Intermediate Value Theorem (2.4) Xiannan Li Kansas State University January 29th, 2017 Intuitive Definition: A function f(x) is continuous at a if you can draw the graph of y = f(x) without

More information

ter. on Can we get a still better result? Yes, by making the rectangles still smaller. As we make the rectangles smaller and smaller, the

ter. on Can we get a still better result? Yes, by making the rectangles still smaller. As we make the rectangles smaller and smaller, the Area and Tangent Problem Calculus is motivated by two main problems. The first is the area problem. It is a well known result that the area of a rectangle with length l and width w is given by A = wl.

More information

Calculus I. 1. Limits and Continuity

Calculus I. 1. Limits and Continuity 2301107 Calculus I 1. Limits and Continuity Outline 1.1. Limits 1.1.1 Motivation:Tangent 1.1.2 Limit of a function 1.1.3 Limit laws 1.1.4 Mathematical definition of a it 1.1.5 Infinite it 1.1. Continuity

More information

Practice Test - Chapter 2

Practice Test - Chapter 2 Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several

More information

Practice Test - Chapter 2

Practice Test - Chapter 2 Graph and analyze each function. Describe the domain, range, intercepts, end behavior, continuity, and where the function is increasing or decreasing. 1. f (x) = 0.25x 3 Evaluate the function for several

More information

Limits and Their Properties

Limits and Their Properties Chapter 1 Limits and Their Properties Course Number Section 1.1 A Preview of Calculus Objective: In this lesson you learned how calculus compares with precalculus. I. What is Calculus? (Pages 42 44) Calculus

More information

Do now as a warm up: Is there some number a, such that this limit exists? If so, find the value of a and find the limit. If not, explain why not.

Do now as a warm up: Is there some number a, such that this limit exists? If so, find the value of a and find the limit. If not, explain why not. Do now as a warm up: Is there some number a, such that this limit exists? If so, find the value of a and find the limit. If not, explain why not. 1 Continuity and One Sided Limits To say that a function

More information

MATH 113: ELEMENTARY CALCULUS

MATH 113: ELEMENTARY CALCULUS MATH 3: ELEMENTARY CALCULUS Please check www.ualberta.ca/ zhiyongz for notes updation! 6. Rates of Change and Limits A fundamental philosophical truth is that everything changes. In physics, the change

More information

Infinite Limits. By Tuesday J. Johnson

Infinite Limits. By Tuesday J. Johnson Infinite Limits By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Graphing functions Working with inequalities Working with absolute values Trigonometric

More information

GUIDED NOTES 5.6 RATIONAL FUNCTIONS

GUIDED NOTES 5.6 RATIONAL FUNCTIONS GUIDED NOTES 5.6 RATIONAL FUNCTIONS LEARNING OBJECTIVES In this section, you will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identify

More information

Limits, Continuity, and the Derivative

Limits, Continuity, and the Derivative Unit #2 : Limits, Continuity, and the Derivative Goals: Study and define continuity Review limits Introduce the derivative as the limit of a difference quotient Discuss the derivative as a rate of change

More information

Continuity at a Point

Continuity at a Point Continuity at a Point When we eplored the limit of f() as approaches c, the emphasis was on the function values close to = c rather than what happens to the function at = c. We will now consider the following

More information

Math Section Bekki George: 08/28/18. University of Houston. Bekki George (UH) Math /28/18 1 / 37

Math Section Bekki George: 08/28/18. University of Houston. Bekki George (UH) Math /28/18 1 / 37 Math 1431 Section 14616 Bekki George: bekki@math.uh.edu University of Houston 08/28/18 Bekki George (UH) Math 1431 08/28/18 1 / 37 Office Hours: Tuesdays and Thursdays 12:30-2pm (also available by appointment)

More information

2.1 The Tangent and Velocity Problems

2.1 The Tangent and Velocity Problems 2.1 The Tangent and Velocity Problems Tangents What is a tangent? Tangent lines and Secant lines Estimating slopes from discrete data: Example: 1. A tank holds 1000 gallons of water, which drains from

More information

Continuity and One-Sided Limits. By Tuesday J. Johnson

Continuity and One-Sided Limits. By Tuesday J. Johnson Continuity and One-Sided Limits By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Rationalizing numerators and/or denominators Trigonometric skills reviews

More information

Math 473: Practice Problems for Test 1, Fall 2011, SOLUTIONS

Math 473: Practice Problems for Test 1, Fall 2011, SOLUTIONS Math 473: Practice Problems for Test 1, Fall 011, SOLUTIONS Show your work: 1. (a) Compute the Taylor polynomials P n (x) for f(x) = sin x and x 0 = 0. Solution: Compute f(x) = sin x, f (x) = cos x, f

More information

1.2 Functions and Their Properties Name:

1.2 Functions and Their Properties Name: 1.2 Functions and Their Properties Name: Objectives: Students will be able to represent functions numerically, algebraically, and graphically, determine the domain and range for functions, and analyze

More information

Sect Polynomial and Rational Inequalities

Sect Polynomial and Rational Inequalities 158 Sect 10.2 - Polynomial and Rational Inequalities Concept #1 Solving Inequalities Graphically Definition A Quadratic Inequality is an inequality that can be written in one of the following forms: ax

More information

Things to remember: x n a 1. x + a 0. x n + a n-1. P(x) = a n. Therefore, lim g(x) = 1. EXERCISE 3-2

Things to remember: x n a 1. x + a 0. x n + a n-1. P(x) = a n. Therefore, lim g(x) = 1. EXERCISE 3-2 lim f() = lim (0.8-0.08) = 0, " "!10!10 lim f() = lim 0 = 0.!10!10 Therefore, lim f() = 0.!10 lim g() = lim (0.8 - "!10!10 0.042-3) = 1, " lim g() = lim 1 = 1.!10!0 Therefore, lim g() = 1.!10 EXERCISE

More information

Chapter 1 Functions and Limits

Chapter 1 Functions and Limits Contents Chapter 1 Functions and Limits Motivation to Chapter 1 2 4 Tangent and Velocity Problems 3 4.1 VIDEO - Secant Lines, Average Rate of Change, and Applications......................... 3 4.2 VIDEO

More information

1.10 Continuity Brian E. Veitch

1.10 Continuity Brian E. Veitch 1.10 Continuity Definition 1.5. A function is continuous at x = a if 1. f(a) exists 2. lim x a f(x) exists 3. lim x a f(x) = f(a) If any of these conditions fail, f is discontinuous. Note: From algebra

More information

Section 1.4 Tangents and Velocity

Section 1.4 Tangents and Velocity Math 132 Tangents and Velocity Section 1.4 Section 1.4 Tangents and Velocity Tangent Lines A tangent line to a curve is a line that just touches the curve. In terms of a circle, the definition is very

More information

Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal

Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal Asymptote Example 2: Real-World Example: Use Graphs

More information

Topic 3 Outline. What is a Limit? Calculating Limits Infinite Limits Limits at Infinity Continuity. 1 Limits and Continuity

Topic 3 Outline. What is a Limit? Calculating Limits Infinite Limits Limits at Infinity Continuity. 1 Limits and Continuity Topic 3 Outline 1 Limits and Continuity What is a Limit? Calculating Limits Infinite Limits Limits at Infinity Continuity D. Kalajdzievska (University of Manitoba) Math 1520 Fall 2015 1 / 27 Topic 3 Learning

More information

1 Functions and Graphs

1 Functions and Graphs 1 Functions and Graphs 1.1 Functions Cartesian Coordinate System A Cartesian or rectangular coordinate system is formed by the intersection of a horizontal real number line, usually called the x axis,

More information

MAT01A1: Precise Definition of a Limit and Continuity

MAT01A1: Precise Definition of a Limit and Continuity MAT01A1: Precise Definition of a Limit and Continuity Dr Craig 7 March 2018 Semester Test 1 D1 LAB 110 Be seated by 08h15. Everything up to and including Ch 2.3. Bring student cards. No bags. No calculators.

More information

Making Connections with Rational Functions and Equations

Making Connections with Rational Functions and Equations Section 3.5 Making Connections with Rational Functions and Equations When solving a problem, it's important to read carefully to determine whether a function is being analyzed (Finding key features) or

More information

MTH4100 Calculus I. Lecture notes for Week 4. Thomas Calculus, Sections 2.4 to 2.6. Rainer Klages

MTH4100 Calculus I. Lecture notes for Week 4. Thomas Calculus, Sections 2.4 to 2.6. Rainer Klages MTH4100 Calculus I Lecture notes for Week 4 Thomas Calculus, Sections 2.4 to 2.6 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 One-sided its and its at infinity

More information

9.5. Polynomial and Rational Inequalities. Objectives. Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater.

9.5. Polynomial and Rational Inequalities. Objectives. Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater. Chapter 9 Section 5 9.5 Polynomial and Rational Inequalities Objectives 1 3 Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater. Solve rational inequalities. Objective 1

More information

Determine whether the formula determines y as a function of x. If not, explain. Is there a way to look at a graph and determine if it's a function?

Determine whether the formula determines y as a function of x. If not, explain. Is there a way to look at a graph and determine if it's a function? 1.2 Functions and Their Properties Name: Objectives: Students will be able to represent functions numerically, algebraically, and graphically, determine the domain and range for functions, and analyze

More information

This Week. Professor Christopher Hoffman Math 124

This Week. Professor Christopher Hoffman Math 124 This Week Sections 2.1-2.3,2.5,2.6 First homework due Tuesday night at 11:30 p.m. Average and instantaneous velocity worksheet Tuesday available at http://www.math.washington.edu/ m124/ (under week 2)

More information

MATH 114 Calculus Notes on Chapter 2 (Limits) (pages 60-? in Stewart)

MATH 114 Calculus Notes on Chapter 2 (Limits) (pages 60-? in Stewart) Still under construction. MATH 114 Calculus Notes on Chapter 2 (Limits) (pages 60-? in Stewart) As seen in A Preview of Calculus, the concept of it underlies the various branches of calculus. Hence we

More information

Advanced Mathematics Unit 2 Limits and Continuity

Advanced Mathematics Unit 2 Limits and Continuity Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring

More information

Advanced Mathematics Unit 2 Limits and Continuity

Advanced Mathematics Unit 2 Limits and Continuity Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring

More information

Horizontal and Vertical Asymptotes from section 2.6

Horizontal and Vertical Asymptotes from section 2.6 Horizontal and Vertical Asymptotes from section 2.6 Definition: In either of the cases f(x) = L or f(x) = L we say that the x x horizontal line y = L is a horizontal asymptote of the function f. Note:

More information

Definition (The carefully thought-out calculus version based on limits).

Definition (The carefully thought-out calculus version based on limits). 4.1. Continuity and Graphs Definition 4.1.1 (Intuitive idea used in algebra based on graphing). A function, f, is continuous on the interval (a, b) if the graph of y = f(x) can be drawn over the interval

More information

Jim Lambers MAT 460 Fall Semester Lecture 2 Notes

Jim Lambers MAT 460 Fall Semester Lecture 2 Notes Jim Lambers MAT 460 Fall Semester 2009-10 Lecture 2 Notes These notes correspond to Section 1.1 in the text. Review of Calculus Among the mathematical problems that can be solved using techniques from

More information

Limits at Infinity. Horizontal Asymptotes. Definition (Limits at Infinity) Horizontal Asymptotes

Limits at Infinity. Horizontal Asymptotes. Definition (Limits at Infinity) Horizontal Asymptotes Limits at Infinity If a function f has a domain that is unbounded, that is, one of the endpoints of its domain is ±, we can determine the long term behavior of the function using a it at infinity. Definition

More information

Math 115 Spring 11 Written Homework 10 Solutions

Math 115 Spring 11 Written Homework 10 Solutions Math 5 Spring Written Homework 0 Solutions. For following its, state what indeterminate form the its are in and evaluate the its. (a) 3x 4x 4 x x 8 Solution: This is in indeterminate form 0. Algebraically,

More information

Polynomial and Rational Functions. Chapter 3

Polynomial and Rational Functions. Chapter 3 Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) -30

More information

Section 3.3 Limits Involving Infinity - Asymptotes

Section 3.3 Limits Involving Infinity - Asymptotes 76 Section. Limits Involving Infinity - Asymptotes We begin our discussion with analyzing its as increases or decreases without bound. We will then eplore functions that have its at infinity. Let s consider

More information

Last week we looked at limits generally, and at finding limits using substitution.

Last week we looked at limits generally, and at finding limits using substitution. Math 1314 ONLINE Week 4 Notes Lesson 4 Limits (continued) Last week we looked at limits generally, and at finding limits using substitution. Indeterminate Forms What do you do when substitution gives you

More information

Math 261 Calculus I. Test 1 Study Guide. Name. Decide whether the limit exists. If it exists, find its value. 1) lim x 1. f(x) 2) lim x -1/2 f(x)

Math 261 Calculus I. Test 1 Study Guide. Name. Decide whether the limit exists. If it exists, find its value. 1) lim x 1. f(x) 2) lim x -1/2 f(x) Math 261 Calculus I Test 1 Study Guide Name Decide whether the it exists. If it exists, find its value. 1) x 1 f(x) 2) x -1/2 f(x) Complete the table and use the result to find the indicated it. 3) If

More information

Chapter 1: Limits and Continuity

Chapter 1: Limits and Continuity Chapter 1: Limits and Continuity Winter 2015 Department of Mathematics Hong Kong Baptist University 1/69 1.1 Examples where limits arise Calculus has two basic procedures: differentiation and integration.

More information

The Intermediate Value Theorem If a function f (x) is continuous in the closed interval [ a,b] then [ ]

The Intermediate Value Theorem If a function f (x) is continuous in the closed interval [ a,b] then [ ] Lecture 2 5B Evaluating Limits Limits x ---> a The Intermediate Value Theorem If a function f (x) is continuous in the closed interval [ a,b] then [ ] the y values f (x) must take on every value on the

More information

Exam 1. (2x + 1) 2 9. lim. (rearranging) (x 1 implies x 1, thus x 1 0

Exam 1. (2x + 1) 2 9. lim. (rearranging) (x 1 implies x 1, thus x 1 0 Department of Mathematical Sciences Instructor: Daiva Pucinskaite Calculus I January 28, 2016 Name: Exam 1 1. Evaluate the it x 1 (2x + 1) 2 9. x 1 (2x + 1) 2 9 4x 2 + 4x + 1 9 = 4x 2 + 4x 8 = 4(x 1)(x

More information

Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1),

Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1), Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1), 4.-4.6 1. Find the polynomial function with zeros: -1 (multiplicity ) and 1 (multiplicity ) whose graph passes

More information

The First Derivative Test for Rise and Fall Suppose that a function f has a derivative at every poin x of an interval A. Then

The First Derivative Test for Rise and Fall Suppose that a function f has a derivative at every poin x of an interval A. Then Derivatives - Applications - c CNMiKnO PG - 1 Increasing and Decreasing Functions A function y = f(x) is said to increase throughout an interval A if y increases as x increases. That is, whenever x 2 >

More information

The degree of a function is the highest exponent in the expression

The degree of a function is the highest exponent in the expression L1 1.1 Power Functions Lesson MHF4U Jensen Things to Remember About Functions A relation is a function if for every x-value there is only 1 corresponding y-value. The graph of a relation represents a function

More information

Solutions to Math 41 First Exam October 15, 2013

Solutions to Math 41 First Exam October 15, 2013 Solutions to Math 41 First Exam October 15, 2013 1. (16 points) Find each of the following its, with justification. If the it does not exist, explain why. If there is an infinite it, then explain whether

More information

Section 2: Limits and Continuity

Section 2: Limits and Continuity Chapter 2 The Derivative Business Calculus 79 Section 2: Limits and Continuity In the last section, we saw that as the interval over which we calculated got smaller, the secant slopes approached the tangent

More information

Continuity. The Continuity Equation The equation that defines continuity at a point is called the Continuity Equation.

Continuity. The Continuity Equation The equation that defines continuity at a point is called the Continuity Equation. Continuity A function is continuous at a particular x location when you can draw it through that location without picking up your pencil. To describe this mathematically, we have to use limits. Recall

More information

Section 2.6: Continuity

Section 2.6: Continuity Section 2.6: Continuity Problem 1 (a) Let f(x) = x 1 x 2 5x. Then f(2) = 1 6 and f(6) = 5, but there is no value of c between 2 6 and 6 for which f(c) = 0. Does this fact violate the Intermediate Value

More information

SEE and DISCUSS the pictures on pages in your text. Key picture:

SEE and DISCUSS the pictures on pages in your text. Key picture: Math 6 Notes 1.1 A PREVIEW OF CALCULUS There are main problems in calculus: 1. Finding a tangent line to a curve though a point on the curve.. Finding the area under a curve on some interval. SEE and DISCUSS

More information

Limits Student Study Session

Limits Student Study Session Teacher Notes Limits Student Study Session Solving limits: The vast majority of limits questions can be solved by using one of four techniques: SUBSTITUTING, FACTORING, CONJUGATING, or by INSPECTING A

More information

Definitions & Theorems

Definitions & Theorems Definitions & Theorems Math 147, Fall 2009 December 19, 2010 Contents 1 Logic 2 1.1 Sets.................................................. 2 1.2 The Peano axioms..........................................

More information

Integer-Valued Polynomials

Integer-Valued Polynomials Integer-Valued Polynomials LA Math Circle High School II Dillon Zhi October 11, 2015 1 Introduction Some polynomials take integer values p(x) for all integers x. The obvious examples are the ones where

More information

Midterm Review. Name: Class: Date: ID: A. Short Answer. 1. For each graph, write the equation of a radical function of the form y = a b(x h) + k.

Midterm Review. Name: Class: Date: ID: A. Short Answer. 1. For each graph, write the equation of a radical function of the form y = a b(x h) + k. Name: Class: Date: ID: A Midterm Review Short Answer 1. For each graph, write the equation of a radical function of the form y = a b(x h) + k. a) b) c) 2. Determine the domain and range of each function.

More information

Math 106 Calculus 1 Topics for first exam

Math 106 Calculus 1 Topics for first exam Chapter 2: Limits and Continuity Rates of change and its: Math 06 Calculus Topics for first exam Limit of a function f at a point a = the value the function should take at the point = the value that the

More information

October 27, 2018 MAT186 Week 3 Justin Ko. We use the following notation to describe the limiting behavior of functions.

October 27, 2018 MAT186 Week 3 Justin Ko. We use the following notation to describe the limiting behavior of functions. October 27, 208 MAT86 Week 3 Justin Ko Limits. Intuitive Definitions of Limits We use the following notation to describe the iting behavior of functions.. (Limit of a Function A it is written as f( = L

More information

LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS

LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS RECALL: VERTICAL ASYMPTOTES Remember that for a rational function, vertical asymptotes occur at values of x = a which have infinite its (either positive or

More information

Simplifying Rationals 5.0 Topic: Simplifying Rational Expressions

Simplifying Rationals 5.0 Topic: Simplifying Rational Expressions Simplifying Rationals 5.0 Topic: Simplifying Rational Expressions Date: Objectives: SWBAT (Simplify Rational Expressions) Main Ideas: Assignment: Rational Expression is an expression that can be written

More information

MATH CALCULUS I 1.5: Continuity

MATH CALCULUS I 1.5: Continuity MATH 12002 - CALCULUS I 1.5: Continuity Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 12 Definition of Continuity Intuitively,

More information

Analysis of Functions

Analysis of Functions Lecture for Week 11 (Secs. 5.1 3) Analysis of Functions (We used to call this topic curve sketching, before students could sketch curves by typing formulas into their calculators. It is still important

More information

THS Step By Step Calculus Chapter 1

THS Step By Step Calculus Chapter 1 Name: Class Period: Throughout this packet there will be blanks you are epected to fill in prior to coming to class. This packet follows your Larson Tetbook. Do NOT throw away! Keep in 3 ring binder until

More information

56 CHAPTER 3. POLYNOMIAL FUNCTIONS

56 CHAPTER 3. POLYNOMIAL FUNCTIONS 56 CHAPTER 3. POLYNOMIAL FUNCTIONS Chapter 4 Rational functions and inequalities 4.1 Rational functions Textbook section 4.7 4.1.1 Basic rational functions and asymptotes As a first step towards understanding

More information

V. Graph Sketching and Max-Min Problems

V. Graph Sketching and Max-Min Problems V. Graph Sketching and Max-Min Problems The signs of the first and second derivatives of a function tell us something about the shape of its graph. In this chapter we learn how to find that information.

More information

Continuity. Chapter 4

Continuity. Chapter 4 Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

More information

Holes in a function. Even though the function does not exist at that point, the limit can still obtain that value.

Holes in a function. Even though the function does not exist at that point, the limit can still obtain that value. Holes in a function For rational functions, factor both the numerator and the denominator. If they have a common factor, you can cancel the factor and a zero will exist at that x value. Even though the

More information

Continuity. Chapter 4

Continuity. Chapter 4 Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

More information

Calculus I. George Voutsadakis 1. LSSU Math 151. Lake Superior State University. 1 Mathematics and Computer Science

Calculus I. George Voutsadakis 1. LSSU Math 151. Lake Superior State University. 1 Mathematics and Computer Science Calculus I George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 151 George Voutsadakis (LSSU) Calculus I November 2014 1 / 67 Outline 1 Limits Limits, Rates

More information

2-5 Rational Functions

2-5 Rational Functions 19. SALES The business plan for a new car wash projects that profits in thousands of dollars will be modeled by the function p (z) =, where z is the week of operation and z = 0 represents opening. a. State

More information

Pre-Calculus Mathematics Limit Process Calculus

Pre-Calculus Mathematics Limit Process Calculus NOTES : LIMITS AND DERIVATIVES Name: Date: Period: Mrs. Nguyen s Initial: LESSON.1 THE TANGENT AND VELOCITY PROBLEMS Pre-Calculus Mathematics Limit Process Calculus The type of it that is used to find

More information

1.1 Introduction to Limits

1.1 Introduction to Limits Chapter 1 LIMITS 1.1 Introduction to Limits Why Limit? Suppose that an object steadily moves forward, with s(t) denotes the position at time t. The average speed over the interval [1,2] is The average

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 6 B) 14 C) 10 D) Does not exist

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 6 B) 14 C) 10 D) Does not exist Assn 3.1-3.3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit, if it exists. 1) Find: lim x -1 6x + 5 5x - 6 A) -11 B) - 1 11 C)

More information

Calculus I Practice Test Problems for Chapter 2 Page 1 of 7

Calculus I Practice Test Problems for Chapter 2 Page 1 of 7 Calculus I Practice Test Problems for Chapter Page of 7 This is a set of practice test problems for Chapter This is in no way an inclusive set of problems there can be other types of problems on the actual

More information

2.1 The Tangent and Velocity Problems

2.1 The Tangent and Velocity Problems 2.1 The Tangent and Velocity Problems Ex: When you jump off a swing, where do you go? Ex: Can you approximate this line with another nearby? How would you get a better approximation? Ex: A cardiac monitor

More information