1. Limits and Continuity

Theory and Examples

If limx→−2 f(x) / x² = 1, find

b. limx→−2 f(x) / x

5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

a. log_3(x)

Limits with trigonometric functions

Find the limits in Exercises 43–50.

limx→−π √(x + 4) cos(x + π)

Determine the following limits.

$\underset{z\to 4}{\lim }\frac{z-5}{{(z^2-10z+24)}^2}$

Oblique Asymptotes

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

y = (x² − 4) / (x − 1)

Determine the following limits. 

lim x→−∞ (2x^-8 + 4x³)

Determine the following limits.

c. lim x→1 x / |x − 1|

Determine the following limits.

a. lim x→1^+ x / |x − 1|

Complete the following steps for the given functions. 

c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.

$f\left(x\right)=\frac{x^2-2x+5}{3x-2}$

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.

If limx→a f(x) = L, then f(a)=L.

9. True, or false? As x→∞,

c. x = O(x+5)

Find the limit: $\underset{t\to 7}{lim}\frac{7+t^2}{7-t^2}$

Determine the following limits.

lim x→0^− 2 / tan x

Determine the following limits.

$\underset{\theta \to 0^{-}}{\lim }\frac{\sin \theta }{{\cos }^2\theta -1}$

Determine the following limits.

a. lim x→−2^+ (x − 4) / x(x + 2)