1. Limits and Continuity

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.

Centering Intervals About a Point

In Exercises 1–6, sketch the interval (a,b), on the x-axis with the point c inside. Then find a value of δ>0 such that a < x < b whenever 0 < |x−c| < δ.

a=4/9, b=4/7, c=1/2

Finding Deltas Graphically

In Exercises 7–14, use the graphs to find a δ>0 such that |f(x)−L| <ε whenever 0< |x−c| <δ.

Finding Deltas Algebraically

Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.

f(x) = 1/x, L = 1/4, c = 4, ε = 0.05

Finding Deltas Algebraically

Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.

f(x) = mx, m > 0, L = 2m, c = 2, ε = 0.03

Using the Formal Definition

Each of Exercises 31–36 gives a function f(x), a point c, and a positive number ε. Find L = lim x→c f(x). Then find a number δ > 0 such that |f(x)−L| < ε whenever 0 < |x−c| < δ.

f(x) = −3x − 2, c = −1, ε = 0.03

[Technology Exercise] Grinding engine cylinders Before contracting to grind engine cylinders to a cross-sectional area of 9in², you need to know how much deviation from the ideal cylinder diameter of c = 3.385in. you can allow and still have the area come within 0.01in² of the required 9in². To find out, you let A=π(x/2)² and look for the largest interval in which you must hold x to make |A − 9| ≤ 0.01. What interval do you find?