1. Limits and Continuity

Finding Limits

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

lim x →π cos² (x― tan x)

Determine the following limits.

a. $\underset{x\to 1^{+}}{\lim }\frac{x-3}{\sqrt{x^2-5x+4}}$

Evaluate lim x→2^+ √x−2.

Determine the following limits.

b. lim x→−2^− (x − 4) / x(x + 2)

Find the limit.

$\(\lim\)_{x\(\rarr\)-\(\pi\)}\(\frac{\sin x}{x}\)$

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

f. (e^x)/2

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

lim p→2 3p / √4p + 1 − 1

112. True, or false? Give reasons for your answers.

e. sec^(-1)x = O(1)

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→2^− f(x)

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

f. f(x) = sech(x), g(x) = e^(-x)

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

71. lim (x → (π/2)⁻) sec x / tan x

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

lim x→−95x

Determine the following limits.

lim x→1 √5x+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) = (x² − 4x + 3) / (x − 1)

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all horizontal asymptotes.

_________

/ x² + 9

d. y = / -------------

√ 9x² + 1