1. Limits and Continuity

The hyperbolic cosine function, denoted $\cosh \left(x\right)$, is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $\cosh \left(x\right)=\frac{e^x+e^{-x}}{2}$.

a. Determine its end behavior by analyzing $\underset{x\to \infty }{\lim }\cosh \left(x\right)$ and $\underset{x\to -\infty }{\lim }\cosh \left(x\right)$.

Evaluate each limit. 

lim θ→0 (1/(2+sinθ)-1/2)/sin θ

Evaluate each limit. 

lim x→0 cos x−1 / sin^2x

Evaluate each limit. 

lim x→0 e^4x−1 / e^x−1

Evaluate each limit. 

lim x→0+ 1−cos^2x / sin x

Evaluate each limit. 

lim x→e^2 ln^2x−5 ln x+6 lnx−2

Determine the following limits.

$\underset{t\to \infty }{\lim }\frac{\cos \left(t\right)}{e^{3t}}$

Evaluate $\underset{x\to \infty }{\lim }f\left(x\right)$ and$\underset{x\to -\infty }{\lim }f\left(x\right)$.

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

Evaluate ${\displaystyle\lim_{x\to\infty}{f(x)}}$ and${\displaystyle\lim_{x\to-\infty}{f(x)}}$.

$f\left(x\right)=\frac{6e^x+20}{3e^x+4}$

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.

$f\left(x\right)=\frac{x^2-3}{x+6}$

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.

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

Use analytic methods to find the value of lim x→π/4 cos 2x / cos x − sin x.

Determine the following limits.

lim θ→π/2 sin^2 θ − 5 sin θ + 4 / sin^2 θ − 1

Determine the following limits.

lim x→π/2 1/√sin x − 1 / x + π/2

Evaluate ${\displaystyle\lim_{x\to\infty}{f(x)}}$ and${\displaystyle\lim_{x\to-\infty}{f(x)}}$.

$f\left(x\right)=1-e^{-2x}$