91Ó°ÊÓ

Use the Extreme Value Theorem to show that each function f has both a maximum and a minimum value on [a, b]. Then use a graphing utility to approximate values M and m in [a, b] at which f has a maximum and a minimum, respectively. You may assume that these functions are continuous everywhere.

$f\left(x\right)=3−2x^2+x^3,[a,b]=[0,2]$

$\underset{x→0}{\lim }\left(\frac{2x}{e^x-1}\right)$

Calculate each limit in Exercises 35–80.
$\underset{x→∞}{lim}\frac{x^{-3}}{x^2-x^{-1}}$

Write delta-epsilon proofs for each of the limit statements $\underset{x→c}{\lim }f\left(x\right)=L$in Exercises $47-60$.

$\underset{x→3}{\lim }\left(x^2-6x+11\right)=2$.

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$x∈(c-\mathrm{δ},c)∪(c,c+\mathrm{δ})\mathrm{then}\mathrm{f}\left(\mathrm{x}\right)∈(\mathrm{L}-\mathrm{Î\mu },\mathrm{L}+\mathrm{Î\mu })\mathit{}$

$\underset{x→-1}{lim}\frac{x^2-2x-3}{x+1}=-4,\mathrm{Î\mu }=0.1$

Calculate each limit in Exercises 35–80.
$\underset{x→0^{+}}{lim}\frac{x^{-3}}{x^2-x^{-1}}$

Use the Extreme Value Theorem to show that each function f in Exercises 49–54 has both a maximum and a minimum value on $[a,b]$. Then use a graphing utility to approximate values M and m in $[a,b]$ at which f has a maximum and a minimum, respectively. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=3-2x^2+x^3; \\ [a,b]=[-1,1] \end{gathered}$$

Sketch graphs by hand and use them to make approximations for each of the limits in Exercises 53–66. If a two-sided limit does not exist, describe the one-sided limits.

$\underset{x→-1}{\lim }\left(x^3-2\right)$

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.

$\underset{x→1^{+}}{lim}\frac{1}{x^2-1}=∞,M=1000,\mathrm{find}\mathrm{largest}\mathrm{δ}>0$

$\underset{x→\mathrm{π}}{\lim }\frac{1}{\cos ec(\mathrm{π}-\mathrm{x})}$