1. Limits and Continuity

Given that $d\left(t\right)$ is a known function and $d$ is a specific value, which of the following procedures will find the time $t$ at which $d\left(t\right)=d$?

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.

$f\left(x\right)=\frac{4x^3+4x^2+7x+4}{x^2+1}$

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

lim x→3 −5x / √4x − 3

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

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

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 1) / (2x + 4) as

b. x→−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)=√x^2+2x+6−3 / x−1

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

lim x → -5^- f(x)

9. True, or false? As x→∞,

e. e^x = o(e^(2x))

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

f(x) = (x⁴ − 1)/(x^2−1)

Determine the following limits.

$\underset{x\to 0^{-}}{\lim }\csc \left(x\right)$

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\(\right\))$.

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

Finding One-Sided Limits Algebraically

Find the limits in Exercises 11–20.

limx→1− (1/(x + 1))((x + 6)/x)((3 − x)/7)

Given that $\underset{x\to 1}{lim}f\left(x\right)=1$, find the largest value of $\epsilon >0$ such that if $|x-1|<\delta$, then $\left|f\right(x)-1|<0.2$. Which of the following values of $\delta$ satisfies this condition?

Calculating Limits

Find the limits in Exercises 11–22.

limx→−1/2 4x(3x+4)²

a. Analyze ${\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{f(x)}}$ and${\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{f(x)}}$ for each function.

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