Evaluate the following limit. If the limit does not exist, select 'DNE'.
$$\underset{x\to \infty }{lim}arctan\left(e^x\right)$$

Use a series expansion to evaluate the limit: $\underset{x\to 0}{lim}\frac{(1+x)-1-\frac{1}{2}{x}^2}{x^2}$. What is the value of this limit?

Consider the function $f\left(x\right)=\{\begin{array}{cc}{x}^2& \text{if}x<2\\ 5& \text{if}x=2\\ x+2& \text{if}x>2\end{array}$. Determine if $f\left(x\right)$ has any points of discontinuity, and explain your reasoning.

Estimate the limit $\underset{n\to \infty }{lim}{(2n+1)}^{-9}$ correct to five decimal places.

Given the function $f\left(x\right)=x^2$, find a number $\delta$ > $0$ such that if $|x-1|<\delta$, then $|x^2-1|<\frac{1}{2}$.

Given the graph of a function $f$, find a number $\epsilon >0$ such that if $|x-1|<\epsilon$, then $\left|f\right(x)-1|<0.2$.

Given the graph of $f$, find a number $\delta$ such that if $0<|x-3|<\delta$, then $\left|f\right(x)-2|<0.5$.

Given that $f^{\left(n\right)}\left(0\right)$=$(n+1)!$ for $n=0,1,2,\dots$, which of the following is the Maclaurin series for $f\left(x\right)$?

Given the curve $r\left(t\right)=(3t,4t)$, for $t\ge 0$, starting from the point $t=0$, which of the following is the correct reparametrization of the curve in terms of arclength $s$?