1. Limits and Continuity

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 $\theta =\frac{\pi }{2}$?

Given that the definite integral from $0$ to $\pi$ of ${\sin }^4\left(x\right)$ $dx$ equals $\frac{3\pi }{8}$, what is the value of the definite integral from $0$ to $\pi$ of ${\sin }^4(\pi -x)$ $dx$?

Given the double integral $\int {\int }_{y=0}{y=2}^{}{\int }_{x=y^2}{x=4}^{}f(x,y) dx dy$, which of the following represents the correct limits of integration after changing the order of integration?

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx →4 (9 − x) = 5

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 

$f\left(x\right)=\sin x$

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

Which of the following regions in the plane has an area equal to the limit $\underset{n\to \infty }{lim}\frac{1}{n}(1+\frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}+\dots +\frac{1}{n^{n-1}})$? Do not evaluate the limit.

Given the geometric series $\sum n_0=\infty x^n$, find the sum of the series $\sum n_1=\infty n{x}^{n-1}$, where $\left|x\right|<1$.