91Ó°ÊÓ

For each limit $\underset{x→c}{lim}f\left(x\right)=L$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\mathrm{δ}$such that if $x∈(c-\mathrm{δ},c)∪(c,c+\mathrm{δ})\mathrm{then}\mathrm{f}\left(\mathrm{x}\right)∈(\mathrm{L}-\mathrm{Î\mu },\mathrm{L}+\mathrm{Î\mu })\mathit{}$.

role="math" localid="1648027918805" $\underset{\mathrm{x}→2}{\lim }{\mathrm{x}}^3=8,\mathrm{Î\mu }=0.5$

In Exercises 39–44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

$f\left(x\right)=\left\{\begin{array}{c}x+1,ifx<1\\ 3x-1,if1â\copyright ½x<2\\ x+2,ifxâ\copyright ¾2\end{array}\right..$

Calculate each of the limits in Exercises $29-70$.

$\underset{x→1}{\lim }\frac{x+x^2-2x^3}{x-x^2}$.

For each limit statement in Exercises $41-44$, use algebra to find $\mathrm{δ}$or $N$in terms of $\mathrm{Î\mu }$or $M$, according to the appropriate formal limit definition.

$\underset{x→1^{-}}{\lim }\frac{1}{1-x}=∞$, find$\mathrm{δ}$in terms of$M$.

Use tables of values to make educated guesses for each of the limits in Exercises 39–52.

$\underset{x→3}{\lim }\frac{x-3}{\left(x^2-2\right)(x-3)}$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$1^{∞}$.

Use tables of values to make educated guesses for each of the limits in Exercises 39–52.

$\underset{x→5}{\lim }\frac{x-5}{x^2-25}$

Indeterminate forms: Identify which of the limit forms listed here are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$0^{-∞}$.

Calculate each limit in Exercises 35–80.

$\underset{x→2}{lim}\frac{x+1}{{(x-2)}^2}$

For each limit $\underset{x→c}{lim}f\left(x\right)=L$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\mathrm{δ}$such that if $x∈(c-\mathrm{δ},c)∪(c,c+\mathrm{δ})\mathrm{then}\mathrm{f}\left(\mathrm{x}\right)∈(\mathrm{L}-\mathrm{Î\mu },\mathrm{L}+\mathrm{Î\mu })\mathit{}$.

$\underset{\mathrm{x}→2}{\lim }{\mathrm{x}}^3=8,\mathrm{Î\mu }=0.25$