1. Limits and Continuity

Determine the following limits.

a. lim x→2^+ x^2 − 4x + 3 / (x − 2)^2

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=|1−x^2| / x(x+1)

Determine the following limits. 

lim x→−∞ 40x^4+x^2+5x / √64x^8+x^6

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=3e^x+10 / e^x

Using Limit Rules

Suppose lim x→0 f(x) = 1 and lim x→0 g(x) = −5. Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation.

limx→0 (2f(x) − g(x)) / (f(x) + 7)² = limx→0 (2f(x) − g(x)) / limx→0 (f(x) + 7)² (a)

(We assume the denominator is nonzero.)

(lim x→0 2f(x) − lim x→0 g(x)) / (lim x→0 (f(x) + 7))² (b)

= (2 lim x→0 f(x) − lim x→0 g(x)) / (lim x→0 f(x) + lim x→0 7)² (c)

= ((2)(1) − (−5)) / (1 + 7)² = 7/64

Finding Limits

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

lim h →0 ((x + h)² ― x²)/h

Determine the following limits.

lim x→−∞ e^x sin x

Assume lim x→1 f(x)=8,lim x→1 g(x)=3, and lim x→1 h(x)=2 Compute the following limits and state the limit laws used to justify your computations.

lim x→1 (4f(x))

A sine limit It can be shown that 1−x^2/ 6 ≤ sin x/ x ≤1, for x near 0.

Use these inequalities to evaluate lim x→0 sin x/ x.

Suppose limx→4 f(x) = 0 and lim x→4 g(x) = −3. Find

c. limx→4 (g(x))²

Find the limit.

$\(\lim\)_{x\(\rarr\)3}\(\frac{\sqrt{x}\)-\(\sqrt\)3}{x-3}$

82. Use the definitions of the hyperbolic functions to find each of the following limits.

i. lim(x→-∞) csch x

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

f(x)=|1−x^2| / x(x+1)

Suppose that limx→−2 p(x) = 4, limx→−2 r(x) = 0, and limx→−2 s(x) = −3. Find

a. limx→−2 (p(x) + r(x) + s(x))

109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.

e. f(x) = arccsc(x), g(x) = 1/x