1. Limits and Continuity

Given the function $f\left(x\right)=x^2$, what is the average rate of change of $f\left(x\right)$ on the interval $(-1,2)$?

Which of the following best describes the remainder estimate for the integral test when determining the convergence of a series $\sum \underset{\infty }{n=1}{a}_n$ with $a_n=f\left(n\right)$, where $f$ is positive, continuous, and decreasing for $x\ge 1$?

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

Which of the following functions is $o\left(x^2\right)$ as $x$ approaches $0$?

Which of the following series can be shown to converge by using the ratio test?

Which of the following best describes the difference between the average rate of change and the instantaneous rate of change of a function $f\left(x\right)$ at a point $x=a$?

Given the graph of $f$, find a number $\delta$ such that if $0<|x-3|<\delta$, then $\left|f\right(x)-2|<0.5$.

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 $\underset{n\to \infty }{lim}{a}_n=8$?