1. Limits and Continuity

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

Given the function $f\left(x\right)=2x^2-3x+1$, which of the following is the average rate of change of $f\left(x\right)$ over the interval from $x=-6$ to $x=-3$?

Find the exact length of the curve $y=x-x^2+{\sin -1}^x$ from $x=0$ to $x=1$.

Assume lim x→1 f(x)=8,lim x→1 g(x)=3, and lim x→1 h(x)=2 Compute the following limits and state the limit laws used to justify your computations.

lim x→1 f(x) / g(x)−h(x)

Determine the following limits.

lim x→1 x^3 − 7x^2 + 12x / 4 − x

Which of the following is the correct algebraic method to find the limit $\underset{x\to 2}{lim}\frac{x^2-4}{x-2}$?

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

How many tangent lines to the curve $y=\frac{x}{x+1}$ pass through the point $(1,\frac{1}{2})$?

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

At what point do the curves $r_1\left(t\right)=(t,2-t,35+t^2)$ and $r_2\left(s\right)=(7-s,s-5,s^2)$ intersect?

Find the exact length of the curve $y=2\sqrt{3}{x}^{\frac{3}{2}}$ for $0\le x\le 4$.

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

Given that $d\left(t\right)$ is a known function and $d$ is a specific value, which of the following procedures will find the time $t$ at which $d\left(t\right)=d$?

Given that $\underset{x\to 1}{lim}f\left(x\right)=1$, find the largest value of $\epsilon >0$ such that if $|x-1|<\delta$, then $\left|f\right(x)-1|<0.2$. Which of the following values of $\delta$ satisfies this condition?