1. Limits and Continuity

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.

lim ( 1 / x²/³ + 2 / (x − 1)²/³ ) as

a. x → 0⁺

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

d. ${\lim }_{x\to 1}f\left(x\right)$

Determine $\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\))$ and $\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\))$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{40x^5+x^2}{16x^4-2x}$

Formal Definitions of One-Sided Limits

Greatest integer function Find (a) limx→400+ ⌊x⌋ and (b) limx→400− ⌊x⌋; then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about limx→400 ⌊x⌋? Give reasons for your answer.

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (x² − x + sin x) / 2x

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).

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

h(x)=e^x(x+1)^3

Find the limit using the graph of $f\(\left\)(x\(\right\))$shown.

$\(\lim\)_{x\(\rarr\)4}f\(\left\)(x\(\right\))$

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.

Use an appropriate limit definition to prove the following limits.

lim x→1 (5x−2) =3;

Use the precise definition of infinite limits to prove the following limits.

$\underset{x\to 4}{\lim }\frac{1}{{(x-4)}^2}=\infty$

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→−3 (x² − 9) / (x + 3) = −6

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→−5⁻ (3x) / (2x + 10)

Theory and Examples

Suppose that f is an odd function of x. Does knowing that limx→0+ f(x) = 3 tell you anything about limx→0− f(x)? Give reasons for your answer.

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)

lim x → ±∞ k(x) = 1, lim x → 1⁻ k(x) = ∞, and lim x → 1⁺ k(x) = −∞