Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes

Similar documents
Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes

and lim lim 6. The Squeeze Theorem

Review: Limits of Functions - 10/7/16

Calculus I Exam 1 Review Fall 2016

2.1 Limits, Rates of Change and Slopes of Tangent Lines

CH 2: Limits and Derivatives

= lim. (1 + h) 1 = lim. = lim. = lim = 1 2. lim

MAT 210 Test #1 Solutions, Form A

Absolute and Local Extrema

Math 106 Calculus 1 Topics for first exam

1.5 Inverse Trigonometric Functions

Limits at Infinity. Use algebraic techniques to help with indeterminate forms of ± Use substitutions to evaluate limits of compositions of functions.

Math 2250 Exam #3 Practice Problem Solutions 1. Determine the absolute maximum and minimum values of the function f(x) = lim.

L Hopital s Rule. We will use our knowledge of derivatives in order to evaluate limits that produce indeterminate forms.

Section 2.5. Evaluating Limits Algebraically

Infinite Limits. Infinite Limits. Infinite Limits. Previously, we discussed the limits of rational functions with the indeterminate form 0/0.

Written Homework 7 Solutions

(e) 2 (f) 2. (c) + (d). Limits at Infinity. 2.5) 9-14,25-34,41-43,46-47,56-57, (c) (d) 2

Infinite Limits. By Tuesday J. Johnson

Concepts of graphs of functions:

3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23

This Week. Professor Christopher Hoffman Math 124

AP Calculus Summer Prep

1 DL3. Infinite Limits and Limits at Infinity

MATH 1910 Limits Numerically and Graphically Introduction to Limits does not exist DNE DOES does not Finding Limits Numerically

2.1 The Tangent and Velocity Problems

L Hopital s Rule. We will use our knowledge of derivatives in order to evaluate limits that produce indeterminate forms.

2. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large. lim

INTERMEDIATE VALUE THEOREM

Math Calculus f. Business and Management - Worksheet 12. Solutions for Worksheet 12 - Limits as x approaches infinity

1) If f x symmetric about what? (Box in one:) (2 points) the x-axis the y-axis the origin none of these

Topics from Algebra and Pre-Calculus. (Key contains solved problems)

DRAFT - Math 101 Lecture Note - Dr. Said Algarni

(c) Find the equation of the degree 3 polynomial that has the same y-value, slope, curvature, and third derivative as ln(x + 1) at x = 0.

MTH30 Review Sheet. y = g(x) BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE

Aim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x)

Calculus II Lecture Notes

MAT137 Calculus! Lecture 20

UNIT 3. Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions

3.3 Limits and Infinity

Limits Involving Infinity (Horizontal and Vertical Asymptotes Revisited)

2.1 The Tangent and Velocity Problems

Solutions to Math 41 First Exam October 15, 2013

Precalculus Honors Problem Set: Elementary Trigonometry

Limits and Continuity

UNIT 3. Recall From Unit 2 Rational Functions

3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:

Constructing Taylor Series

Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations.

Updated: January 16, 2016 Calculus II 7.4. Math 230. Calculus II. Brian Veitch Fall 2015 Northern Illinois University

MA1021 Calculus I B Term, Sign:

Advanced Mathematics Unit 2 Limits and Continuity

Advanced Mathematics Unit 2 Limits and Continuity

Mathematic 108, Fall 2015: Solutions to assignment #7

Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued)

Chapter 5: Integrals

ICM ~ Unit 4 ~ Day 3. Horizontal Asymptotes, End Behavior

Review Problems for Test 1

Section 1.x: The Variety of Asymptotic Experiences

Solutions to Math 41 First Exam October 18, 2012

AP Calculus BC Prerequisite Knowledge

AP Calculus. Limits, Continuity, and Differentiability

Rational Functions. A rational function is a function that is a ratio of 2 polynomials (in reduced form), e.g.

f(x) g(x) = [f (x)g(x) dx + f(x)g (x)dx

Section 1.4 Tangents and Velocity

The above statement is the false product rule! The correct product rule gives g (x) = 3x 4 cos x+ 12x 3 sin x. for all angles θ.

AP Calculus ---Notecards 1 20

1.4 Techniques of Integration

Chapter 1: Limits and Continuity

Introduction to Determining Power Law Relationships

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

Integration by Substitution

Unit 1 PreCalculus Review & Limits

8-5. A rational inequality is an inequality that contains one or more rational expressions. x x 6. 3 by using a graph and a table.

Set 3: Limits of functions:

Chapter 5: Limits, Continuity, and Differentiability

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

Using the Derivative. Chapter Local Max and Mins. Definition. Let x = c be in the domain of f(x).

MATH 31B: MIDTERM 2 REVIEW. sin 2 x = 1 cos(2x) dx = x 2 sin(2x) 4. + C = x 2. dx = x sin(2x) + C = x sin x cos x

Calculus I Sample Exam #01

LIMITS AND DERIVATIVES

Topics and Concepts. 1. Limits

Chapter 13: General Solutions to Homogeneous Linear Differential Equations

Horizontal asymptotes

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

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)

CALCULUS AB SUMMER ASSIGNMENT

Name: AK-Nummer: Ergänzungsprüfung January 29, 2016

Arnie Pizer Rochester Problem Library Fall 2005 WeBWorK assignment LimitsRates0Theory due 01/01/2006 at 02:00am EST.

Assignment 13 Assigned Mon Oct 4

Calculus. Weijiu Liu. Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA

Directions: Please read questions carefully. It is recommended that you do the Short Answer Section prior to doing the Multiple Choice.

AP Calculus AB Chapter 1 Limits

Inverse Trig Functions

Horizontal asymptotes

Limit. Chapter Introduction

Announcements. Topics: Homework:

4.4 Graphs of Logarithmic Functions

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

A.P. Calculus BC Test Two Section One Multiple-Choice Calculators Allowed Time 40 minutes Number of Questions 15

Transcription:

Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes

Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes

Infinite Limits Ex : Find,, and by x 0 + x x 0 x x 0 x a) making a table b) looking at a graph

Infinite Limits Ex : Find,, and by x 0 + x x 0 x x 0 x a) making a table x -0. -0.0-0.00 0 0.00 0.0 0. /x -0-00 -000? 000 00 0 So x 0 + x = + and x 0 x = x 0 x = DNE

Infinite Limits Ex : Find,, and by x 0 + x x 0 x x 0 x b) looking at a graph x 0 + x = + x 0 x = x 0 x = DNE

Infinite Limits Ex 2: Find,, and by x 0 + x 2 a) making a table b) looking at a graph x 0 x 2 x 0 x 2

Infinite Limits Ex 2: Find,, and by x 0 + x 2 a) making a table x 0 x 2 x 0 x 2 x -0. -0.0-0.00 0 0.00 0.0 0. /x^2 00 0,000,000,000?,000,000 0,000 00 So x 0 + x 2 = + and x 0 x 2 = + x 0 x 2 = +

Infinite Limits Ex 2: Find,, and by x 0 + x 2 b) looking at a graph x 0 x 2 x 0 x 2 x 0 + x 2 = + x 0 x 2 = + x 0 x 2 = +

Informal Definition of an Infinite Limit ) f x = + means x a if you plug in numbers closer and closer to a (on either side of a but not a itself!) into the function, the outputs will get bigger and bigger without bound. 2) f x = + means x a if you plug in numbers closer and closer to a but less than a into the function, the outputs will get bigger and bigger without bound. 3) f x = + means + x a If you plug in numbers closer and closer to a but greater than a into the function, the outputs will get bigger and bigger without bound.

Informal Definition of an Infinite Limit ) f x = means x a if you plug in numbers closer and closer to a (on either side of a but not a itself!) into the function, the outputs will get smaller and smaller without bound. 2) f x = means x a if you plug in numbers closer and closer to a but less than a into the function, the outputs will get smaller and smaller without bound. 3) f x = means + x a If you plug in numbers closer and closer to a but greater than a into the function, the outputs will get smaller and smaller without bound.

Infinite Limits Notes: When the it is of the form nonzero, the answer for the 0 it will either be +,, or DNE. Q: How can you tell which one? A: Think about making a table and consider the sign of the outputs. ) If the outputs are always positive (eventually), the answer will be + 2) If the outputs are always negative (eventually), the answer will be 3) Otherwise the answer will be DNE

Infinite Limits Notes: In this class, if a one-sided it is of the form nonzero the answer for the it will either be + or (not DNE). The only time we ll see DNE is if the left and right hand its are different. If a it is + or, this doesn t mean that the it exists. The it doesn t exist because if a it exists, it must be a number. But by saying that a it is + or, you are explaining HOW it doesn t exist. But if a it is + or, do not write that the it DNE, just know that it doesn t exist. 0,

Infinite Limits Notes: Think of a it of the form nonzero 0 like this: When you divide (a nonzero number) by a really small number, you get a really big number (+ or )

Ex 3: Find x 4 2x 9 x 4 7 Infinite Limits

Infinite Limits Ex 4 (book sec. 2.2 hw #39): Find xcsc(x) x 2π

Infinite Limits Ex 5 (book sec. 2.2 hw #42): Find x 0 + x lnx

Infinite Limits Ex 6: Find x 2 lnx 2x

Vertical Asymptotes Q: What is a vertical asymptote?

Vertical Asymptotes Q: What is a vertical asymptote? vertical asymptotes x = a & x = b

Definition of a Vertical Asymptotes Def: The line x = a is a vertical asymptote of the function f if x a f x =, f x = +, x a f x = or f x = + x a + x a +

Vertical Asymptotes Steps in finding vertical asymptotes Find the points (a) of discontinuity of f Calculate f x and f x x a x a + If either of these answers are + or, then the line x = a is a vertical asymptote. Otherwise, the line x = a is not a vertical asymptote.

Vertical Asymptotes Ex 7: Find the vertical asymptotes of f x = x2 + 5x 4 a) x 2 b) g x = 7x + 0 sin (x) x c) h x = ln (x) d) k x = sin x e) L x = 2x 0 x 5

Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes

Limits at Infinity Ex 8: Find x x by a) making a table b) looking at a graph

Limits at Infinity Ex 8: Find x x by a) making a table -,000,000-0,000-00 -0.00000-0.000-0.0 So x x = 0

Limits at Infinity Ex 8: Find x x by b) looking at the graph So x x = 0

Limits at Infinity Ex 9: Find x x 2 by a) Making a table b) Looking at a graph

Limits at Infinity Ex 9: Find x x 2 by a) Making a table 00 0,000,000,000 0.000 0^-8 0^-2 So x x 2 = 0

Limits at Infinity Ex 9: Find x x 2 by b) looking at the graph So x x 2 = 0

Limits at Infinity Ex 0: Use a table to find x 8x + 2x 5

Limits at Infinity Ex 0: Use a table to find x 8x + 2x 5 0 00 000 5.4 4.07692 4.00526

Limits at Infinity Ex 0: Use a table to find x 8x + 2x 5 0 00 000 5.4 4.07692 4.00526 So guess is x 8x + 2x 5 = 4

Limits at Infinity Ex : Use the graph to find each of the following a) x tan (x) b) c) x ln (x) sin(x) x

Definition of a Limit at Infinity / Horizontal Asymptotes Let L be a real number and f be a function. ) f x = L means x if you plug in bigger and bigger (positive) numbers into the function, the outputs will get closer and closer to L. 2) If f x = L, then the line y = L is called a x horizontal asymptote of f.

Definition of a Limit at Infinity/ 3) f x = L means x Horizontal Asymptotes if you plug in smaller and smaller (or bigger and bigger negative) numbers into the function, the outputs will get closer and closer to L. 4) If f x = L, then the line y = L is called a x horizontal asymptote of f.

Definition of a Limit at Infinity/ Horizontal Asymptotes Steps in finding the horizontal asymptotes of a function Find f x. If this it exists and equals the y x number L, then the line = L is a horizontal asymptote of the function f. Find f x. If this it exists and equals the y x number R, then the line = R is a horizontal asymptote of the function f.

Definition of a Limit at Infinity/ Horizontal Asymptotes Notes: A function can have 0,, or 2 horizontal asymptotes Rational expressions can have 0 or horizontal asymptotes, but not 2 (as you ll see shortly) When finding the horizontal asymptotes of algebraic expressions, be careful with your algebra.

Result: ) If r > 0 is a rational number, then x x r = 0 2) If r > 0 is a rational number such that is defined for all x, then x x r = 0 x r

Horizontal Asymptotes Ex 2: Find the horizontal asymptotes of a) f x = x + 3 x 2 7x + 4 b) g x = 5x3 + 4x 9 2x 3 + x + c) h x = 2x3 + 3 x 2

Result on Horizontal Asymptotes of Rational Expressions If the degree top < degree bottom, the graph only has one horizontal asymptote. The horizontal asymptote is y = 0 (i.e. the x-axis) If the degree top = degree bottom, the graph only has one horizontal asymptote. The horizontal asymptote is lead coefficient of top y = lead coefficient of bottom If the degree top > degree bottom, the graph has no horizontal asymptote

Horizontal Asymptotes Ex 3: Find the horizontal asymptotes of a) f x = 5x 3 4x 6 + 2x 4 3 b) g x = 27x6 4x 2 + 3 3 2x 3 8x

Limits at Infinity Ex 4: Find x 4x2 + 3x + 2x

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback