1. Limits and Continuity

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

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

a. Analyze ${\displaystyle\lim_{x\to\infty}{f(x)}}$ and${\displaystyle\lim_{x\to-\infty}{f(x)}}$ for each function.

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

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

$\underset{x\to \infty }{\lim }x\cos \left(x\right)$

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

$\underset{x\to 0}{\lim }\frac{1-\cos \left(x\right)}{{\cos }^2\left(x\right)-3\cos \left(x\right)+2}$

Evaluate each limit. 

$\underset{x\to \pi }{\lim }\frac{{\cos }^2\left(x\right)+3\cos \left(x\right)+2}{{\cos }^{}\left(x\right)+1}$

Evaluate each limit. 

$\underset{x\to 3}{\lim }\sqrt{x^2+7}$

Evaluate each limit. 

$\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin \left(x\right)-1}{\sqrt{\sin \left(x\right)}-1}$