1. Limits and Continuity

Find all vertical asymptotes $x=a$ of the following functions. For each value of $a$, determine $\underset{x\to a^{+}}{\lim }f\left(x\right)$, $\underset{x\to a^{-}}{\lim }f\left(x\right)$, and $\underset{x\to a}{\lim }f\left(x\right)$.

$f\left(x\right)=\frac{\cos \left(x\right)}{x^2+2x}$

Find all vertical asymptotes $x=a$ of the following functions. For each value of $a$, determine ${\displaystyle\lim_{x\to a^{+}}}f\left(x\right)$, ${\displaystyle\lim_{x\to a^{-}}}f\left(x\right)$, and ${\displaystyle\lim_{x\to a}}f\left(x\right)$.

$f\left(x\right)=\frac{x+1}{x^3-4x^2+4x}$

Explain the meaning of lim x→a f(x) =∞.

Use the graph of f(x) = x / (x² − 2x − 3)² to determine lim x→−1 f(x) and lim x→3 f(x).

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 − 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