1. Limits and Continuity

Analyze the following limits and find the vertical asymptotes of f(x) =(x − 5) / (x² − 25).

lim x→−5⁺ f(x)

Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x⁴ − 49x²).

lim x → -7 f(x) 

Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x⁴ − 49x²).

lim x→0 f(x)

Use the definitions given in Exercise 57 to prove the following infinite limits.

lim x→1^- 1 / 1 − x=∞

Evaluate lim x→0 x + 1/ 1 −cos x.

Sketch a possible graph of a function g, together with vertical asymptotes, satisfying all the following conditions.

g(2) =1,g(5) =−1,lim x→4 g(x) =−∞,lim x→7^− g(x) =∞,lim x→7^+ g(x) =−∞

{Use of Tech} Population growth Consider the following population functions.

d. Evaluate and interpret lim t→∞ p(t).

p(t) = 600 (t²+3/t²+9)

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 sin ax / sin bx, where a and b are constants with b ≠ 0.

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 3x) / x

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 7x) / 3x

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 5x) / x

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 7x) / (sin x)

Use Theorem 3.10 to evaluate the following limits.

lim x🠂2 (sin (x-2)) / (x² - 4)

How is lim x🠂0 sin x/x used in this section?

Another method for proving lim x→0 cos x−1/x = 0 Use the half-angle formula sin²x = 1− cos 2x/2 to prove that lim x→0 cos x−1/x=0.