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?

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

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).

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

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

$\lim_{x\rightarrow0^{-}}f\left(x\right)$ , $\lim_{x\rightarrow0^{+}}f\left(x\right)$, $\lim_{x\rightarrow0}f\left(x\right)$

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

$\lim_{x\rightarrow4^{-}}f\left(x\right)$, $\lim_{x\rightarrow4^{+}}f\left(x\right)$, $\lim_{x\rightarrow4}f\left(x\right)$

Find the specified limit or state that the limit does not exist by creating a table of values.

$f\left(x\right)=\frac{1}{x}$

$\lim_{x\rightarrow1^{-}}f\left(x\right)$, $\lim_{x\rightarrow1^{+}}f\left(x\right)$, $\lim_{x\rightarrow1}f\left(x\right)$

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

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