91Ó°ÊÓ

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-x-2} $$ $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & \\ \hline \end{array} $$

Decide whether the problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning and use a graphical or numerical approach to estimate the solution. A bicyclist is riding on a path modeled by the function \(f(x)=0.08 x\), where \(x\) and \(f(x)\) are measured in miles. Find the rate of change of elevation when \(x=2\).

Discuss the continuity of the function on the closed interval. $$ g(x)=\sqrt{25-x^{2}} \quad[-5,5] $$

Discuss the continuity of the function on the closed interval. $$ \text { Function } \quad \text { Interval } $$ $$ g(x)=\frac{1}{x^{2}-4} \quad[-1,2] $$

Find the limit \(L\). Then find \(\delta>0\) such that \(|f(x)-L|<0.01\) whenever \(0<|x-c|<\delta\). $$ \lim _{x \rightarrow 5}\left(x^{2}+4\right) $$

Does the graph of every rational function have a vertical asymptote? Explain.

Find the limit (if it exists). $$ \lim _{x \rightarrow 0} \frac{[1 /(3+x)]-(1 / 3)}{x} $$

A jeweler resizes a ring so that its inner circumference is 6 centimeters. (a) What is the radius of the ring? (b) If the ring's inner circumference can vary between 5.5 centimeters and \(6.5\) centimeters, how can the radius vary? (c) Use the \(\varepsilon-\delta\) definition of limit to describe this situation. Identify \(\varepsilon\) and \(\delta\).

A sporting goods manufacturer designs a golf ball having a volume of \(2.48\) cubic inches. (a) What is the radius of the golf ball? (b) If the ball's volume can vary between \(2.45\) cubic inches and \(2.51\) cubic inches, how can the radius vary? (c) Use the \(\varepsilon-\delta\) definition of limit to describe this situation. Identify \(\varepsilon\) and \(\delta\).

A crossed belt connects a 20-centimeter pulley ( \(10-\mathrm{cm}\) radius) on an electric motor with a 40 -centimeter pulley \((20-\mathrm{cm}\) radius) on a saw arbor (see figure). The electric motor runs at 1700 revolutions per minute. (a) Determine the number of revolutions per minute of the saw. (b) How does crossing the belt affect the saw in relation to the motor? (c) Let \(L\) be the total length of the belt. Write \(L\) as a function of \(\phi\), where \(\phi\) is measured in radians. What is the domain of the function? (Hint: Add the lengths of the straight sections of the belt and the length of the belt around each pulley.) (d) Use a graphing utility to complete the table. $$\begin{array}{|l|l|l|l|l|l|}\hline \boldsymbol{\phi} & 0.3 & 0.6 & 0.9 & 1.2 & 1.5 \\\\\hline \boldsymbol{L} & & & & & \\ \hline\end{array}$$ (e) Use a graphing utility to graph the function over the appropriate domain. (f) Find \(\lim _{\phi \rightarrow(\pi / 2)^{-}} L\). Use a geometric argument as the basis of a second method of finding this limit. (g) Find \(\lim _{\phi \rightarrow 0^{+}} L\).