Finding Limits

In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)

g(x) = 1/(2 + (1/x))

Finding Limits

In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)

f(x) = 2/x − 3

Limits of quotients

Find the limits in Exercises 23–42.

limx→1 (x −1) / (√(x + 3) − 2)

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

Limits and Infinity

Find the limits in Exercises 37–46.

sin x

lim ------------- ( If you have a grapher, try graphing

x→∞ |x| the function for ―5 ≤ x ≤ 5 ) .

Finding Limits of Differences When x → ±∞

Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)

lim x → ∞ (√(x + 9) − √(x + 4))

At what points are the functions in Exercises 13–30 continuous?

y = 1/(|x| + 1) − x²/2

Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find

b. limx→c 2f(x)g(x)

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→1 f(x) = 1 if f(x) = {x², x ≠ 1

2, x = 1

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→0 (1 + x + sin x) / (3 cosx)

At what points are the functions in Exercises 13–30 continuous?

y = cos (x) / x

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→9 √(x − 5) = 2

Find the limits in Exercises 31–40. Are the functions continuous at the point being approached?

lim x → 0 tan (π/4 cos (sin x¹/³))

Calculating Limits

Find the limits in Exercises 11–22.

limx→−1/2 4x(3x+4)²

Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = 4 + 3x² / (x² + 1)

Removable discontinuity Give an example of a function f (x) that is continuous for all values of x except x = 2, where it has a removable discontinuity. Explain how you know that f is discontinuous at x = 2, and how you know the discontinuity is removable.

If f(1)=5, must limx→1 f(x) exist? If it does, then must limx→1 f(x)=5? Can we conclude anything about limx→1 f(x)? Explain.

Limits with trigonometric functions

Find the limits in Exercises 43–50.

lim x→0 tan x

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

llimx→0 (x −x cos x) / sin² 3x

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 tan θ / θ²cot 3θ

Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

y = (x + 3)/(x + 2)

Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x → ∞ √((8x² − 3) / (2x² + x))

Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

y = 1/(x − 1)

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.

Limits and Continuity

Graph the function

1 , x ≤ ―1

―x , ―1 < x < 0

ƒ(x) = { 1 , x = 0 ,

―x , 0 < x < 1

1 , x ≥ 1

Then discuss, in detail, limits, one-sided limits, continuity, and one-sided continuity of ƒ at x = ―1 , 0 , and 1. Are any of the discontinuities removable? Explain.

Limits and Continuity

Repeat the instructions of Exercise 1 for

1 , x ≤ ―1

1/x , 0 < |x| < 1

ƒ(x) = { 0, x = 1 ,

1 , x > 1 .

Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–20.

a. limx→1+ (√2x (x − 1)) / |x − 1|

Average Rates of Change

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

g(x)=x²−2x

a. [1, 3]

Limits and Continuity

On what intervals are the following functions continuous?

a. ƒ(x) = tan x

Theory and Examples

a. If limx→0 f(x) / x² = 1, find limx→0 f(x).