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 Methods A) x 2 9 Look for Patterns B) x 3 + 27
C) 8x 3-64
II. Functions, Graphs, and Limits Analysis of graphs. With the aid of technology. Prelude to the use of calculus both to predict and to explain the observed local and global behaviour of a function. 9
Analysis of Graphs Using graphing technology: 1. Sketch the graph of y = x 3 27 10
Analysis of Graphs 1. y = x 3 27 A) Find the zeros x = 3 B) Find the local max and min points These are points that have either the largest, or smallest y value in a particular region, or neighbourhood on the graph. There are no local max or min points 11
C) Identify any points where concavity changes from concave up to concave down (or vice a versa). The point of inflection is (0, -27) 12
2. Sketch the graph of: A) y 2 x 1 x 2 B) y = x 2 What do you notice? y = x 2 is a slant (or oblique) asymptote. 13
Rational Functions f(x) is a rational function if f( x) where p(x) and q(x) are polynomials and qx ( ) 0 px ( ) qx ( ) Rational functions often approach either slant or horizontal asymptotes for large (or small) values of x
Rational Functions are not continuous graphs. There various types of discontinuities. There vertical asymptotes which occur when only the denominator (bottom) is zero. There are holes in the graph when there is zero/zero 0 0
3. Describe what happens to the x 2 function y near x = 2. The graph seems to approach the point (2, 4) What occurs at x = 2? Division by zero. The function is undefined when x = 2. In fact we get There is a hole in the graph. What occurs at x = -2? x 2 4 Division by zero however this time there is a vertical asymptote. 0 0 16
4. Describe what happens to the sinx function y as x gets close to x 0. The function seems to approach 1 Does it make any difference if the calculator is in degrees or radians? Yes, it only approaches 1 in radians. 17
Limits of functions (including one-sided limits). A basic understanding of the limiting process. Estimating limits from graphs or tables of data. Calculating limits using algebra. Calculating limits at infinity and infinite limits
Zeno s Paradox Half of Halves 1 1 1 1 1... 2 4 8 16 32 Mathematically speaking: nn 1 lim i n i 1 2 i 1 i This is the limit of an infinite series 19
How many sides does a circle have? 5 sides? 18 sides? http://www.mathopenref.com/circleareaderive.html 20
Limit of a Function The limit of a function tells how a function behaves near a certain x- value. Suppose if I wanted to go to a certain place in Canada. We would use a map 21
Consider: If we have a function y = f(x) and we are trying to find out what the value of the function is for a x- value under the shaded area, we could make an estimate of what it would be by looking at the function before it goes into or leaves the shaded area. Guess what the function value is at x = 3 22
The smaller the shaded area can be made, the better the approximation would be. Guess what the function value is at x = 3 23
Guess what the function value is at x = 3 24
Guess what the function value is at x = 3 25
Mathematically speaking: As x gets close to a, f(x) gets close to a value L This can be written: lim f ( x ) L xa Note: This is not multiplication. It means The limit of f(x) as x approaches a equals L 26
We can get values of f(x) to be arbitrarily close to L by looking at values of x sufficiently close to a, but not equal to a. It does not matter if f(a) is defined. We are only looking to see what happens to f(x) as x approaches a 27
Limits using a table of values. 1. Determine the behaviour of f (x) as x approaches 2. 28
This is the limit from the right side of x = 2 This is the limit from the left side of x = 2 Examples: (Using a Table of Values) 2 2 x 4 x 4 lim lim 4 x 2 x 2 x 2 x 2 2 2 x x 4 x 4 x 2. Find: 3 2.5 2.1 2.01 2.001 x 2 5 1 4.5 1.5 4.1 1.9 4.01 1.99 4.001 1.999 x 2 3 3.5 3.9 3.99 3.999 2 x 4 lim 4 x 2 x 2 2 x 4 lim 4 x 2 x 2 29
Examples: (Using a Table of Values) sin sin lim lim 1 0 0 2.Find: (radians) 0.1 0.01 0.001 0 sin 0.998334 0.9999998 sin lim 1 (radians) -0.1-0.01-0.001 0 sin 0.998334 0.9999833 0.9999833 0.9999998 sin lim 1 30
3. For the function f 1 ( x ) x, complete the table below x -5-1 1 1 0 1 1 1 5 (x) Sketch the graph of y = f(x) 2 4 y 4 2 x 31
Using the table and graph as a guide, answer following questions: What value is f (x) approaching as x becomes a larger positive number? What value is f (x) approaching as x becomes a larger negative number? Will the value of f (x) ever equal zero? Explain your reasoning. 32
With reference to the previous graph complete the following table 33
One Sided Limits Consider the function below: This is a piecewise function It consists of two different functions combined together into one function What is the equation? f( x) 2 x x 1, 1 x 1, x 1 34
A) B) C) Find the following using the graph and function rule lim f( x) x 1 lim f( x) x 0 lim f( x) x 2 D) lim f( x) x 1 For this limit we need to find both the left and right hand limits because the function has different rules on either side of 1. 35
lim ( ) f x x 1 2 lim x 1 x 1 In this case we say that the limit Does Not Exist (DNE) lim ( ) f x x 1 lim x 1 x 1 = 0 = 2 NOTE: Limits do not exist if the left and right limits at a x-value are different. 36
Mathematically Speaking A function will have a limit L as x approaches a, if and only if as x approaches a from the left and a from the right you get the same value, L. OR: lim f ( x ) xa L ( iff ) lim f ( x ) L xa and lim f ( x ) L xa 37
2.A) Draw f( x) x 2, x 2 2 x 1, x 2 B) Find: lim f( x) x 2 38
3.A) Draw f( x) 2 x x 2, 1 x 2, x 1 B) Find: lim f( x) x 0 C) Find: lim f( x) x 1 39
4. Find x 1 lim f ( x ) where f ( x ) x 2, x 1 2, x 1 40
5. Find lim f ( x ) where f ( x ) x 3 2 x x 4, 3 x 4, x 3 41
Evaluate the limits using the following piecewise function: 42
Identify which limit statements are true and which are false for the graph shown. 43
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Absolute Values Definition: The absolute value of a, a, is the distance a is from zero on a number line. 3 = -3 = x = 2 a a a if a a if a 0 0 Note: - a is positive if a is negative
EX. -5 Here the value is negative so -5 = -(-5) = 5
Rewrite the following without 1. absolute values symbols. 3 2. 3 3. x
4. x + 2 x 2 if x 2 0 ( x 2) if x 2 0 x 2 if x 2 ( x 2) if x 2
5. x = 3 x 3 if x 0 x 3 if x 0 6. x < 3
7. x > 3
Find Recall: x lim x x 0 x0 x0 x, if x 0 x x, if x 0 x x lim lim 1 x x x x lim lim 1 x x x0 x0 x lim x 0 x DNE 51
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Find x 1 lim x 1 x 1 Find x 2 lim 2 x 4 x 2 53
x Greatest Integer Function x is the greatest integer function. It gives the greatest integer that is less than or equal to x. Example: A) 2 B) C) D) 2.2 2.99 0.2 54
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Find lim x 2 x lim x 2 x lim x 2 x lim x 2 x DNE 56
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HERE 59
Solving Limits Using Algebra There are 7 limit laws which basically allow you to do direct substitution when finding limits. Examples: Evaluate and justify each step by indicating the appropriate Limit Law 1. lim 2x 1 x 3 ( 60
2. lim x( x 1) x 0 3. 2 lim2x 3x 1 x 1 61
4. x 3 2x lim x 2 5 x 2 NOTE: : Direct substitution works in many cases, so you should always try it first. 62
NOTE: These limit laws basically allow you to do Direct Substitution. 4. x 3 2x lim x 2 5 x 2 Direct Substitution works in many cases, so you should always try it first. 63
However, there are a few cases (mostly in math courses) where direct substitution does not work immediately, or at all. 64
A) Draw the graph of y f( x) x 2, x 3 2, x 3 x B) Find lim f( x) x 3 In this case direct substitution would give an answer of which is not correct. Remember the limit shows what the function is approaching as x approaches a value. It does not matter what the actual function value is at that x value. 65
1. lim x 1 x Examples Direct substitution gives is undefined. which In this case the limit will not work because the x value the limit is approaching is not in the domain of the function. 1 lim x 1 ( x DNE Does Not Exist 66
Examples 1 2. lim 2 x 0 x 1 Direct substitution gives which is undefined. 0 In this case direct substitution will not work because the x value the limit is approaching is not in the domain of the function. However, as we will see later this one would not be DNE. Here we say that: lim x 0 1 x 2 67
3. lim x 2 2 x 4 x 2 2 2 4 0 2 2 0 Direct Substitution Whenever you get 0, this means there is some simplification you can do to the function before you do the direct substitution. 0 What would you do here?? lim 2 x x x x2 x2 Factor 4 ( 2)( 2) lim x 2 x 2 lim x 2 2 2 4 x 2 68
4. x 2 x 2 lim 2 x 1 2 x x 1 2 1 1 2 2 2(1) 1 1 0 0 Direct Substitution What would you do here?? Factor 69
5. lim h 0 ( h 2 2 4 h ( 2 2 0 4 0 0 0 Direct Substitution What would you do here?? More work!! 70
6. lim x 2 x 2 2 x 2 2 2 4 2 2 0 0 Direct Substitution What would you do here?? Rationalize the Numerator How do we rationalize a square root? We multiply top and bottom by the conjugate. The conjugate is the other factor of the difference of squares 71
lim x 2 x 2 2 x 2 72
7. lim h 0 ( 2 h 1 h 1 2 ( 2 0 1 1 1 1 What would you do here?? 2 2 2 0 0 0 0 Simplify the rational expression 73
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8. Find lim x 4 ( x 2( x 2 x 4 x 4 75
9. Find x 1 x x x 3 x 3 lim x 1 76
lim 10. Find x 2 3x 6 x 2 x 2 77
A) lim x 3 2 x 2x 3 2 x x 6 Practice: 78
B) lim x 1 x x 3 2 1 1 Practice: 79
C) x 9 9 lim 3 Practice: x x 80
D) x 13 4 lim x 3 7 x 2 Practice: 81
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Continuity
What is meant by a continuous function? A curve that can be drawn without taking your pencil from the paper. Which letters of the alphabet are the result of continuous lines?
What functions are continuous? Polynomials These are continuous everywhere f( x) gx ( ) These are continuous for all values of x except for the roots of g(x) = 0. Rational Functions In other words it is continuous for all values in the domain
Exponential and Logarithmic Functions Sine and Cosine graphs Absolute Value Graphs
What type of discontinuities are there?
We need a way of defining continuity to know whether or not a function is discontinuous or continuous at a point. Definition: A function y = f(x) is continuous at a number b, if lim f ( x ) f ( b) xb
This can be broken into 3 parts 1. f(b) is defined (It exists) b is in the domain of f(x) 2. lim f( x) exists. x lim ( ) lim f ( x ) b In other words f x x b x b 3. Part 1 = Part 2 lim f ( x ) f ( b) xb
Describe why each place was discontinuous
Discuss the continuity of the following 1. f(x) = x 3 + 2x + 1 2. gx ( ) 3. hx ( ) x 3 This is continuous everywhere because it is a polynomial. x x 1 Discontinuous at x = 1 (VA) 1 is not in the Domain x 4, x 3 2, x 3 Not continuous at x = 3. WHY? lim f ( x ) 3 4 7 f (3) 2
1, x 1 x x 1, x 1 2 4. f ( x ) x, 1 x 1 x = -1 We need to check x = -1 and x = 1. Do we need to check x = 0? NO! In 1/x, x=0 is not in x < -1 lim f( x) lim x 1 x 1 lim f( x) x 1 lim x 1 1 x Thus f(x) is continuous at x = 1 x 2 1 1
1, x 1 x x 1, x 1 2 4. f ( x ) x, 1 x 1 x = 1 lim ( ) f x x 1 lim x 1 x 2 1 lim f( x) x 1 x 1 lim x 1 2 Thus f(x) is discontinuous at x = 1 since the left and right limits are not the same.
5. y = sinx Continuous everywhere 6. y = cos x Not continuous at VA x k, k 2 7. y = 2 x Continuous everywhere
Examples What value of k would make the following functions continuous? 1. 2 x 4, x 2 f( x) x 2 kx, 2
2. hx ( ) 2 x x x 2, 2 5 x k, x 2
3. f( x) 2 x kx x kx, 1 3, x 1
4. For what value of the constant c is the function f( x) x c, x 2 2 cx 1, x 2 continuous at every number?
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There is one other type of discontinuity Graph 1 y sin x This is known as an Oscillating Discontinuity y x
The function sin(1/x) is not defined at x = 0 so it is not continuous at x = 0. The function also oscillates between -1 and 1 as x approaches 0. Therefore, the limit does not exist.