1. Limits and Continuity

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?

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.

lim x→∞ cot^−1

Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x→⁻∞ (³√x − ⁵√x) / (³√x + ⁵√x)

Find the limit by creating a table of values.

$\lim_{x\rarr0}-4x+2$

Find the limit by creating a table of values.

$\lim_{x\rarr2}3x^2+5x+1$

Find the limit by creating a table of values.

$\lim_{x\rarr1}\frac{x^2-4}{x-2}$

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

$\lim_{x\rarr1}f\left(x\right)$

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

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

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

$\lim_{x\rarr4}f\left(x\right)$

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

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