91Ó°ÊÓ

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.

$\underset{x→∞}{\lim }\frac{x-4}{x^2-4}$

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.

$$\begin{gathered} f\left(x\right)=x^3-3x^2-2; \\ [a,b]=[0,2]; \\ K=-4 \end{gathered}$$

Calculate each limit in Exercises 35–80.

$\underset{x→-∞}{lim}\frac{3^x-5^x}{4^x}$

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

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

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.

$\underset{x→∞}{lim}\ln x=∞,M=100,\mathrm{find}\mathrm{smallest}N>0$

Write delta-epsilon proofs for each of the limit statements in Exercises $47–60.$

$\underset{x→5^{+}}{\lim }\sqrt{x-5}=0$

Calculate each of the limits:

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

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.

$$\begin{gathered} f\left(x\right)=x^3-3x^2-2; \\ [a,b]=[2,4]; \\ K=-4 \end{gathered}$$

Interesting trigonometric limits: For each of the functions that follow, use a calculator or other graphing utility to examine the graph of f near $x=0$. Does it appear that f is continuous at $x=0$? Make sure your calculator is set to radian mode

The function $f$is continuous at$x=c.$

If$\underset{x→-1^{+}}{\lim }f\left(x\right)=-∞$and$\underset{x→-1^{-}}{\lim }f\left(x\right)=-∞$, what can you say about$\underset{x→-1}{\lim }f\left(x\right)$?