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


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

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.

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

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. 

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

Find the horizontal asymptotes of each function using limits at infinity.

f(x) = (2e^x + 3) / (e^x + 1)