Copyright © Cengage Learning. All rights reserved. 1 Limits and Their Properties Copyright © Cengage Learning. All rights reserved.

Limits involving Infinity Day 2 BC Limits Review Limits involving Infinity Handout: (Introduction to Limits (Graphical)WS) 1.4B-1.5 & 3.5 Copyright © Cengage Learning. All rights reserved.

One-Sided Limits and Continuity on a Closed Interval

One-Sided Limits and Continuity on a Closed Interval Figure 1.31

Example 4 – Continuity on a Closed Interval Discuss the continuity of f(x) = Solution: The domain of f is the closed interval [–1, 1].

Example 4 – Solution Because and you can conclude that f is cont’d Because and you can conclude that f is continuous on the closed interval [–1, 1], as shown in Figure 1.32. Figure 1.32

Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. A function is continuous at a point if the limit is the same as the value of the function. 1 2 3 4 This function has discontinuities at x=1 and x=2. It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

Properties of Continuity

Properties of Continuity Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous.

Example – Applying Properties of Continuity By Theorem 1.11, it follows that each of the functions below is continuous at every point in its domain.

The Intermediate Value Theorem (IVT)

The Intermediate Value Theorem In other words

Intermediate Value Theorem If a function is continuous between a and b, then it takes on every value between and . Because the function is continuous, it must take on every y value between and .

The Intermediate Value Theorem The Intermediate Value Theorem tells you that at least one number c exists, but it does not provide a method for finding c. Such theorems are called existence theorems. The Intermediate Value Theorem states that for a continuous function f, if x takes on all values between a and b, f(x) must take on all values between f(a) and f(b).

The Intermediate Value Theorem Suppose that a girl is 5 feet tall on her thirteenth birthday and 5 feet 7 inches tall on her fourteenth birthday. Then, for any height h between 5 feet and 5 feet 7 inches, there must have been a time t when her height was exactly h. This seems reasonable because human growth is continuous and a person’s height does not abruptly change from one value to another.

The Intermediate Value Theorem The Intermediate Value Theorem guarantees the existence of at least one number c in the closed interval [a, b] . There may, of course, be more than one number c such that f(c) = k, as shown in Figure 1.35. Figure 1.35

The Intermediate Value Theorem A function that is not continuous does not necessarily exhibit the intermediate value property. For example, the graph of the function shown in Figure 1.36 jumps over the horizontal line given by y = k, and for this function there is no value of c in [a, b] such that f(c) = k. Figure 1.36

The Intermediate Value Theorem The Intermediate Value Theorem often can be used to locate the zeros of a function that is continuous on a closed interval. Specifically, if f is continuous on [a, b] and f(a) and f(b) differ in sign, the Intermediate Value Theorem guarantees the existence of at least one zero of f in the closed interval [a, b] .

Note that f is continuous on the closed interval [0, 1]. Because Example – An Application of the Intermediate Value Theorem Use the Intermediate Value Theorem to show that the polynomial function has a zero in the interval [0, 1]. Solution: Note that f is continuous on the closed interval [0, 1]. Because it follows that f(x) must pass through zero in the interval [0,2].

Solution cont’d You can therefore apply the Intermediate Value Theorem to conclude that there must be some c in [0, 1] such that as shown in Figure 1.37. Figure 1.37

AP Example: Verify that the Intermediate Value Theorem applies on the interval and find the value of c guaranteed by the theorem.

You try an another AP Example: Explain why the function has a zero in the given interval. f(x) is continuous f(0) = -2 and f(1) = 2 Therefore, by the Intermediate Value Theorem (IVT), f(x)=0 for at least one value of c between 0 and 1.

Discuss with your neighbor Name the three conditions that must be met for a function to be continuous at a point.

1.5 Infinite Limits

Infinite Limits: As the denominator approaches zero, the value of the fraction gets very large. vertical asymptote at x=0. If the denominator is positive then the fraction is positive. If the denominator is negative then the fraction is negative.

Limits at infinity: vertical asymptote at x=0. What happens to the value of this expression as the denominator approaches either positive or negative infinity?

Example – Determining Infinite Limits from a Graph Determine the limit of each function shown in Figure 1.41 as x approaches 1 from the left and from the right. Figure 1.41

Example 1(a) – Solution When x approaches 1 from the left or the right, (x – 1)2 is a small positive number. Thus, the quotient 1/(x – 1)2 is a large positive number and f(x) approaches infinity from each side of x = 1. So, you can conclude that Figure 1.41(a) confirms this analysis. Figure 1.41(a)

Example 1(b) – Solution cont’d When x approaches 1 from the left, x – 1 is a small negative number. Thus, the quotient –1/(x – 1) is a large positive number and f(x) approaches infinity from left of x = 1. So, you can conclude that When x approaches 1 from the right, x – 1 is a small positive number.

Example 1(b) – Solution cont’d Thus, the quotient –1/(x – 1) is a large negative number and f(x) approaches negative infinity from the right of x = 1. So, you can conclude that Figure 1.41(b) confirms this analysis. What is the limit of this function at x=1? DNE Figure 1.41(b)

Example 2 – Finding Vertical Asymptotes Determine all vertical asymptotes of the graph of each function.

Example 2(a) – Solution When x = –1, the denominator of is 0 and the numerator is not 0. So, by Theorem 1.14, you can conclude that x = –1 is a vertical asymptote, as shown in Figure 1.42(a). Figure 1.42(a).

Example 2(b) – Solution By factoring the denominator as cont’d By factoring the denominator as you can see that the denominator is 0 at x = –1 and x = 1. Moreover, because the numerator is not 0 at these two points, you can apply Theorem 1.14 to conclude that the graph of f has two vertical asymptotes, as shown in figure 1.42(b). Figure 1.42(b)

Example 2(c) – Solution By writing the cotangent function in the form cont’d By writing the cotangent function in the form you can apply Theorem 1.14 to conclude that vertical asymptotes occur at all values of x such that sin x = 0 and cos x ≠ 0, as shown in Figure 1.42(c). So, the graph of this function has infinitely many vertical asymptotes. These asymptotes occur at x = nπ, where n is an integer. Figure 1.42(c).

You try:  

You try: Find the limit: a) b) c)

You try:  

Limits at Infinity This section discusses the “end behavior” of a function on an infinite interval. Consider the graph of as shown in Figure 3.33. Figure 3.33

Limits at Infinity Graphically, you can see that the values of f(x) appear to approach 3 as x increases without bound or decreases without bound. You can come to the same conclusions numerically, as shown in the table.

Limits at Infinity The table suggests that the value of f(x) approaches 3 as x increases without bound . Similarly, f(x) approaches 3 as x decreases without bound . These limits at infinity are denoted by and

Horizontal Asymptotes Limits at infinity are horizontal asymptotes. For rational functions, use horizontal asymptote rules.

Example – Finding a Limit at Infinity This number becomes insignificant as . There is a horizontal asymptote at 1.

Same Example – Algebraic Solution There is a horizontal asymptote at 1.

Find: Example: When we graph this function, the limit appears to be zero. so for : by the sandwich theorem:

Example: Find:

Example – Finding a Limit at Infinity Find the limit: Solution: Using Theorem 3.10, you can write

Example – Finding a Limit at Infinity Find the limit: Solution:

Example – Finding a Limit at Infinity

Example – Finding a Limit at Infinity

Example – A Function with Two Horizontal Asymptotes Find each limit.

Example 4 – Solution The graph of is shown in figure 3.38. cont’d

Often you can just “think through” limits. p

Example – Finding Infinite Limits at Infinity Find each limit. Solution: As x increases without bound, x3 also increases without bound. So, you can write As x decreases without bound, x3 also decreases without bound. So, you can write

Example – Solution cont’d The graph of f(x) = x3 in Figure 3.42 illustrates these two results. These results agree with the Leading Coefficient Test for polynomial functions. Figure 3.42

Discuss with your group

You are in Calculus, therefore you rock! Theorem 1.1 You are in Calculus, therefore you rock!

Find the following limit: HWQ Find the following limit:

Homework MMM pgs. 30-31 + 76,77,83,84 pg. 80 Larson

Find the following limit: HWQ Find the following limit:

Find the following limit: HWQ Find the following limit: