Evaluate the limit, if it exists: $\underset{(x,y)\to (0,0)}{lim}\frac{xy}{x^2+y^2}$.

Find the arc length of the graph of the function $y=\frac{2}{3}{x}^{\frac{3}{2}}+\frac{1}{3}$ over the interval $0\le x\le 9$.

Find the limit: $\underset{t\to 7}{lim}\frac{7+t^2}{7-t^2}$

Given the function $f\left(x\right)$ defined as follows:

$f\left(x\right)=\left\{\begin{array}{ll}2x+1& ,x<1\\ 3& ,x=1\\ x^2& ,x>1\end{array}\right.$

Find $\underset{x\to 1}{lim}f\left(x\right)$. (If an answer does not exist, enter 'dne.')

Find the limit: $\underset{t\to 5}{lim}\frac{5+t^2}{5-t^2}$

Find the exact length of the curve $y=\ln (\sec (x\left)\right)$ for $0\le x\le \frac{\pi }{4}$.

Evaluate the following limit, if it exists. If the limit does not exist, select 'DNE'.
$$\underset{(x,y)\to (0,0)}{lim}\frac{xy}{x^2+y^2}$$

Find the exact length of the curve given by $x=2\sqrt{{\sqrt{t}}^3}$, $y=t^2-2$ for $0\le t\le 4$.

Find the limit.

$\lim_{x\rarr0}x^3+5x^2-7x+3$