1. Limits and Continuity

Sketch a graph of y=2^x and carefully draw three secant lines connecting the points P(0, 1) and Q(x,2^x), for x=−3,−2, and −1.

Evaluate ${\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)}}$ and${\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{f(x)}}$.

$f\left(x\right)=1-e^{-2x}$

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→−2^− h(x)

Oblique Asymptotes

Graph the rational functions in Exercises 103–108. Include the graphs and equations of the asymptotes.

y = (x² − 1) / x

Which of the following statements are correct? Choose all that apply.

a. lim x→1 1/ (x−1)^2 does not exist

b. lim x→1 1/ (x−1)^2=∞

c. lim x→1 1/(x−1)^2=−∞

Finding Limits

In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.

lim (4g(x))¹/³ = 2

x →0

Find the limit.

$\(\lim\)_{x\(\rarr\)-2}\(\frac{x^2-5x-14}{x+2}\)$

2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?

e. e^(-x)

Limits and Continuity

Suppose the functions ƒ(x) and g(x) are defined for all x and that lim (x → 0) ƒ(x) = 1/2 and lim (x → 0) g(x) = √2. Find the limits as x → 0 of the following functions.

f. [ƒ(x) • cos x ] / x―1

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

13. lim(x → 1⁻)arcsin(x)

Find the limit.

$\(\lim\)_{x\(\rarr\)3}\(\frac{x^2+2x-3}{x-2}\)$

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.

lim x→a (x² ― a²)/(x⁴ ― a⁴)

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→2 (5x−6)^3/2

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=√x^2+2x+6−3 / x−1

Evaluate each limit and justify your answer. 

lim x→0 (x / √16x+1-1)^1/3