1. Limits and Continuity

Average Rates of Change

In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.

g(t)=2+cos t

b. [0,π]

Average Rates of Change

In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.

P(θ)=θ³ − 4θ² + 5θ; [1,2]

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

g. limx→1 f(x) does not exist.

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

h. f(0)=0

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

i. f(0)=1

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

a. limx→2 f(x) does not exist.

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

b. limx→2 f(x)=2

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

c. limx→1 f(x) does not exist.

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

d. limx→c f(x) exists at every point c in (-1,1).

Using the Sandwich Theorem

a. It can be shown that the inequalities 1 − x²/ 6 < (x sin x) / (2−2cos x) < 1 hold for all values of x close to zero (except for x = 0). What, if anything, does this tell you about limx→0 (x sin x) / (2 − 2cos x)?

Give reasons for your answer.

[Technology Exercise] b. Graph y = 1 − (x²/6), y=(x sinx)/(2 − 2cos x), and y = 1 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.

Using the Sandwich Theorem

a. Suppose that the inequalities 1/2 − x² / 24 < (1 − cos x)/ x² < 1/2 hold for values of x close to zero, except for x = 0 itself. (They do, as you will see in Section 9.9.) What, if anything, does this tell you about limx→0 (1 −cos x)/ x²?

Give reasons for your answer.

[Technology Exercise] b. Graph the equations y=(1/2) − (x²/24), y = (1 - cos x) / x², and y = 1/2 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let f(x) = (x² - 9) / (x + 3)

b. Support your conclusions in part (a) by graphing f near c = -3 and using Zoom and Trace to estimate y-values on the graph as x → −3.

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let g(x) = (x² − 2) / (x − √2)

a. Make a table of the values of g at the points x=1.4,1.41,1.414, and so on through successive decimal approximations of √2. Estimate limx→√2 g(x).

Theory and Examples

If x⁴ ≤ f(x) ≤ x² for x in [−1,1] and x² ≤ f(x) ≤ x⁴ for x < - 1 and x > 1, at what points c do you automatically know limx→c f(x)? What can you say about the value of the limit at these points?

Theory and Examples

Suppose that g(x) ≤ f(x) ≤ h(x) for all x≠2 and suppose that lim x→2 g(x) = lim x→2 h(x) = −5. Can we conclude anything about the values of f, g, and h at x = 2? Could f(2) = 0? Could limx→2 f(x)=0? Give reasons for your answers.