91Ó°ÊÓ

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

$\underset{x→1}{\lim }\left(2x^2-4x+3\right)=1$.

Use the Intermediate Value Theorem to show that for each function f, interval [a, b], and valueK in Exercises 55– 60, there is some c∈(a, b) for which f(c) = K. Then use a graphing utility to approximate all such values c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=5-x^4; \\ [a,b]=[0,2]; \\ K=0 \end{gathered}$$

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.

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

Calculate each limit in Exercises 35–80.
$\underset{x→0^{+}}{lim}\frac{x^{{\displaystyle \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}-x^{{\displaystyle \raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}{x^2}$

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.

$\underset{x→1^{+}}{lim}\frac{1}{x^2-1}=∞,M=10000,\mathrm{find}\mathrm{largest}\mathrm{δ}>0$

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

$\underset{x→2}{\lim }\left(3x^2-12x+15\right)=3$.

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

Calculate each limit in Exercises 35–80.

$\underset{x→∞}{lim}\frac{x^{{\displaystyle \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}-x^{{\displaystyle \raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}{x^2}$

Use the Intermediate Value Theorem to show that for each function f, interval [a, b], and valueK in Exercises 55– 60, there is some c∈(a, b) for which f(c) = K. Then use a graphing utility to approximate all such values c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=5-x^4; \\ [a,b]=[-2,-1]; \\ K=0 \end{gathered}$$

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.

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