91Ó°ÊÓ

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.

$\underset{x→∞}{lim}(4-x^2)=-∞,M=-100,\mathrm{find}\mathrm{smallest}N>0$

For each of the limit statements in Exercises 61-66, write a $\mathrm{δ}-M,N-\mathrm{Ï\mu }$, or $N-M$ proof, according to the type of limit statement.
$\underset{x→∞}{\lim }\frac{2x-1}{x}=2$

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.

$\underset{x→2}{\lim }f\left(x\right),forf\left(x\right)=\left\{\begin{array}{l}{x}^2,ifx<2\\ 1-3x,ifxâ‰\yen 2\end{array}\right.$

Use the Intermediate Value Theorem to show that for each function fand value K in Exercises 61–66, there must be some ³¦âˆˆR for which f(c) = K. You will have to select an appropriate interval [a, b] to work with. Then find or approximate one such value of c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=\sin x; \\ K=\frac{1}{2} \end{gathered}$$

Calculate each limit in Exercises 35–80.

$\underset{x→3^{+}}{lim}\ln (x^2-9)$

For each of the limit statements in Exercises 61-66, write a $\mathrm{δ}-M,N-\mathrm{Ï\mu }$, or $N-M$ proof, according to the type of limit statement.
$\underset{x→-∞}{\lim }\frac{2x-1}{x}=2$

Calculate each limit in Exercises 35–80.

$\underset{x→0^{+}}{lim}\ln \left(\frac{1}{x}\right)$

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.

$\underset{x→∞}{lim}(4-x^2)=-∞,M=-10000,\mathrm{find}\mathrm{smallest}N>0$

Calculate each of the limits:

$\underset{h→0}{\lim }\frac{{(2+h)}^2-2^2}{h}$.

Use the Intermediate Value Theorem to show that for each function fand value K in Exercises 61–66, there must be some ³¦âˆˆR for which f(c) = K. You will have to select an appropriate interval [a, b] to work with. Then find or approximate one such value of c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=\sin x; \\ K=\frac{\sqrt{3}}{2} \end{gathered}$$