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?

Use series to evaluate the limit: $\underset{x\to 0}{lim}\frac{1-\cos (5x)}{1+5x-e^{5x}}$.

Use series to evaluate the limit: $\underset{x\to 0}{lim}\frac{1-\cos \left(2x\right)}{1+2x-e^{2x}}$.

Use the squeeze theorem to find the limit: $\underset{(x,y)\to (0,0)}{lim}xy-\sin \left(\frac{1}{x^2+y^2}\right)$. What is the value of this limit?

Given the graph of the function $g\left(x\right)$ below, on which interval is $g\left(x\right)$ continuous?

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 $|x^2-1|<\frac{1}{2}$.

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