Evaluate the following limits. lim_{thetarightarrow0}frac{cos^2theta-1}{theta}

Evaluate the following limits.
$\underset{(u,v)\to (8,8)}{lim}\frac{u^{1/3}-v^{1/3}}{u^{2/3}-v^{2/3}}$

Use limit theorems as needed to evaluate each of the following limits; if a limit is real number, find it; if a limit is infinite, find which one; and if a limit does not exist, then write DNE.
${\displaystyle \underset{x\to 2}{lim}\frac{(x+1)(x^2-4)}{x^2-3x+2}}$

Find the limit and discuss the continuity of the function.
${\displaystyle \underset{(x,y)\to (3,1)}{lim}(x^2-2y)}$

Analysis of the voltage in a hairdryer involves terms of the form$\sin (nwt-{90}^{\circ })$, where n is a positive integer, w is the frequency of the voltage, and t is time. Use an identity to simplify this expression.

Find the following limit without using L’Hopital’s rule:
$\underset{x\to 1}{lim}\frac{x^m-1}{x^n-1}$

Limit ${\displaystyle \underset{x\to \frac{\pi }{3}}{lim}\frac{\sin (x-\frac{\pi }{3})}{1-2\cos \left\{x\right\}}}$

I need to find
$\underset{x\to 0}{lim}\frac{e^x\sin x-x(1-x)}{x^3}$