1. Limits and Continuity

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 $\left|x\right|<1$.

Use the ratio test to determine whether the series $\sum _{n=1}^{\infty }\frac{\left(n\right)!}{n^n}$ converges or diverges.

Given the graph of a function $f$, find a number $\epsilon >0$ such that if $|x-1|<\epsilon$, then $\left|f\right(x)-1|<0.2$.

For the function $f\left(x\right)=2x^2+3x$, what is the average rate of change from $x=0$ to $x=5$?

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 x sin (1/x) = 0

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Which of the following statements about the series $\sum \underset{}{n=1}\infty \frac{1}{2^n}$ is true?

Consider the function $f\left(x\right)=\{\begin{array}{cc}{x}^2& \text{if}x<2\\ 5& \text{if}x=2\\ x+2& \text{if}x>2\end{array}$. Determine if $f\left(x\right)$ has any points of discontinuity, and explain your reasoning.

Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = 4 + 3x² / (x² + 1)

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^+ tan x

Suppose f(x) lies in the interval (2, 6). What is the smallest value of ε such that |f (x)−4|<ε?

Given the curve $r\left(t\right)=(3t,4t)$, for $t\ge 0$, starting from the point $t=0$, which of the following is the correct reparametrization of the curve in terms of arclength $s$?

For which values of $p$ does the series $\sum _{n=1}^{\infty }\frac{{(-1)}^{n+1}}{n^p}$ converge?

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→1 x^4=1

For the function $f\left(x\right)=2x$, what is the limit of the average rate of change of $f\left(x\right)$ as $x$ increases from $2$ to $4$?

For which values of $x$ does the series $\sum _{n=1}^{\infty }\frac{x^n}{n}$ converge?