1. Limits and Continuity

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

Approximate the sum of the series $\sum _{n=1}\infty \frac{{-}^{n}6n}{n!}$ correct to four decimal places.

Which of the following functions is continuous on the interval $(0,1)$?

For which positive integers $k$ is the following series convergent? $\sum n_1\infty \frac{{(n!)}^2}{(kn)!}$

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 $f\left(x\right)$?

A tangent line approximation of a function value is an overestimate when the function is:

Find the radius of convergence, $r$, of the series $\sum \underset{}{n=1}\stackrel{}{\infty }\frac{{(-1)}^n{x}^n}{5^n}$.

What does it mean to say that $\underset{n\to \infty }{lim}{a}_n=8$?

Evaluate the limit: $\underset{h\to 0}{lim}\frac{{(2+h)}^2-4}{h}$

Suppose the graph of the function $f$ is shown above. What is $\underset{x\to -1}{lim}f\left(f\right(x\left)\right)$?

For which values of p does the improper integral ${\int }_1^{\infty }{x}^{3p+1} dx$ converge?

Use series to evaluate the limit: $\underset{x\to 0}{lim}\frac{\left(\sin \right(3x)-3x+\frac{9}{2}{x}^3)}{x^5}$

Which of the following explains why a function $f\left(x\right)$ is discontinuous at $x=a$?

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

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