Evaluate each limit. 


lim x→0 e^4x−1 / e^x−1

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=cos x+2√x / √x.

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

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