91Ó°ÊÓ

$\underset{x→0}{\lim }\frac{\sin x}{\cos x}$

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.

$\underset{x→∞}{lim}\frac{3x}{x+1}=3,\mathrm{Î\mu }=0.5,\mathrm{find}\mathrm{smallest}N>0$

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.

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

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

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

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.

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

Calculate each limit in Exercises 35–80.

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

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.

$\underset{x→∞}{lim}\frac{3x}{x+1}=3,\mathrm{Î\mu }=0.1,\mathrm{find}smallestN>0$

Calculate each limit in Exercises 35–80.

$\underset{x→∞}{lim}\frac{2^x-4^{-x}}{3^x}$

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

$\underset{x→2}{\lim }\frac{x^2-3x+2}{x-2}=1$

Calculate each of the limits:

$\underset{x→\frac{\mathrm{π}}{2}}{\lim }\frac{cotx}{\cos x}$.