1. Limits and Continuity

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

Suppose $a_n$ and $b_n$ are sequences with positive terms, and the series $\sum b_n$ is known to be convergent. Which of the following statements is true?

To what number does the series $\sum \underset{}{k=0}\stackrel{}{\infty }{\left(-\frac{1}{2}\right)}^k$ converge?

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?

If a function $f$ is continuous on $(-\infty ,\infty )$, which of the following statements is true about its graph?

Which of the following definite integrals is equal to $\underset{n\to \infty }{lim}\sum \frac{10}{n}(1+\frac{5k}{n}{)}^2(k=1n)$?

Find the exact length of the curve $y=\ln (1-x^2)$ for $0\le x\le \frac{1}{3}$.

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

What is the slope of the tangent line to the polar curve $r=2{\theta }^2$ when $\theta =\frac{\pi }{2}$?