91Ó°ÊÓ

Calculate each limit in Exercises 35–80.

$\underset{x→∞}{lim}\frac{2e^{1.5x}}{3e^{2x}+e^{1.5x}}$

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.

$$\begin{gathered} f\left(x\right)=x^3+2; \\ K=-15 \end{gathered}$$

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.

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

Calculate each of the limits:

$\underset{x→0}{\lim }\frac{1}{se{c}^{-1}x}$.

Calculate each limit in Exercises 35–80.

$\underset{x→∞}{lim}\frac{1-5e^{2x}}{3e^x+4e^{2x}}$

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.

$$\begin{gathered} f\left(x\right)=-2x^2+4; \\ K=0 \end{gathered}$$

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.

$\underset{x→-∞}{lim}{3}^x=0,\mathrm{Î\mu }=\frac{1}{4},\mathrm{find}\mathrm{smallest}-\mathrm{magnitude}N<0$

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

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.

$\underset{x→\mathrm{π}}{\lim }cscx$

Calculate each of the limits:

$\underset{h→0}{\lim }\frac{{(3+h)}^2-3^2}{h}$.