1. Limits and Continuity

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?

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 $4$?

Find the limit using the graph of $f\(\left\)(x\(\right\))$shown.

$\(\lim\)_{x\(\rarr\)4}f\(\left\)(x\(\right\))$

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

Find the limit by creating a table of values.

$\(\lim\)_{x\(\rarr\)0}-4x+2$

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\(\rarr\)0}f\(\left\)(x\(\right\))$

Find the limit using the graph of $f\(\left\)(x\(\right\))$shown.

$\(\lim\)_{x\(\rarr\)-2}f\(\left\)(x\(\right\))$

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\(\rarr\)-2}f\(\left\)(x\(\right\))$

Find the limit by creating a table of values.

$\(\lim\)_{x\(\rarr\)1}\(\frac{x^2-4}{x-2}\)$

Find the limit by creating a table of values.

$\(\lim\)_{x\(\rarr\)2}3x^2+5x+1$

Using the graph, find the specified limit or state that the limit does not exist (DNE).

${\lim }_{x\to -2^{-}}f\left(x\right)$, ${\lim }_{x\to -2^{+}}f\left(x\right)$, ${\lim }_{x\to -2^{}}f\left(x\right)$