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.

$f\left(x\right)=x^4−3x^2−2,[a,b]=[−1,1]$

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.

$f\left(x\right)=3−2x^2+x^3,[a,b]=[−1,2]$

Calculate each limit in Exercises 35–80.

$\underset{x→∞}{\lim }(x^{-\frac{1}{3}}-x^{-\frac{1}{2}})$

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

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

$\underset{x→8}{\lim }\left(3x-11\right)=13$.

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

$\underset{x→∞}{\lim }\sin \left(\frac{1}{x}\right)$

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

$\underset{\mathrm{x}→3}{\lim }\left(2-{\mathrm{x}}^2\right)=-7,\mathrm{Î\mu }=0.001$

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 localid="1648023101818" $x∈(c-\mathrm{δ},c)∪(c,c+\mathrm{δ})\mathrm{then}\mathrm{f}\left(\mathrm{x}\right)∈(\mathrm{L}-\mathrm{Î\mu },\mathrm{L}+\mathrm{Î\mu })\mathit{}$

localid="1648023199049" role="math" $\underset{x→-1}{lim}\frac{x^2-2x-3}{x+1}=-4,\mathrm{Î\mu }=1$

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.

$\underset{x→0}{\lim }\frac{1}{x}$

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

$\underset{x→0}{\lim }\left(3x^2+1\right)=1$.