1. Limits and Continuity

Determine the following limits at infinity.

lim x→∞ (5 + 1/x +10/x^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?

Determine whether the following statements are true and give an explanation or counterexample.

The line x=−1 is a vertical asymptote of the function f(x) =x^2 − 7x + 6 / x^2 − 1.

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

a. Estimate lim x→π/4 cos 2x / cos x − sin x by making a table of values of cos 2x / cos x − sin x for values of x approaching π/4. Round your estimate to four digits.

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 ${\sin }^4(\pi -x)$ $dx$?

Use the graph of $f\(\left\)(x\(\right\))$to estimate the value of the limit or state that it does not exist (DNE).

$\(\lim\)_{x\(\rarr\)1}f\(\left\)(x\(\right\))$

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