Chapter 13

Evaluate each limit. (a) \(\lim _{x \rightarrow 0} \sqrt{4-x^{2}}\) (b) \(\lim _{x \rightarrow 3} \sqrt{4-x^{2}}\) (c) \(\lim _{x \rightarrow 2} \sqrt{4-x^{2}}\)

Evaluate each limit. $$\begin{array}{l} \text { (a) } \lim _{x \rightarrow 2^{-}} \ln (x-2) \\ \text { (b) } \lim _{x \rightarrow 2^{+}} \ln (x-2) \\ \text { (c) } \lim _{x \rightarrow 2} \ln (x-2) \end{array}$$

Evaluate each limit. (a) \(\lim _{x \rightarrow 10} \log x\) (b) \(\lim _{x \rightarrow-1} \log x\) (c) \(\lim _{x \rightarrow 0} \log x\)

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 1} \frac{\ln x^{2}}{\ln x}\)

Find \(f^{\prime}(x)\) using the alternative definition. $$f(x)=-x^{2}+4 x$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \frac{e^{-x}-1}{x}\)

Evaluate each limit. $$\begin{array}{l} \text { (a) } \lim _{x \rightarrow \pi^{-}} \cot x \\ \text { (b) } \lim _{x \rightarrow \pi^{+}} \cot x \\ \text { (c) } \lim _{x \rightarrow \pi} \cot x \end{array}$$

Evaluate each limit. (a) \(\lim _{x \rightarrow e^{3}} \ln |x|\) (b) \(\lim _{x \rightarrow-1} \ln |x|\) (c) \(\lim _{x \rightarrow 0} \ln |x|\)

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0}(x \sin x)\)

Find \(f^{\prime}(x)\) using the alternative definition. $$f(x)=2 x^{2}-x$$

Evaluate each limit. $$\begin{array}{l} \text { (a) } \lim _{x \rightarrow 0^{-}} \log |x| \\ \text { (b) } \lim _{x \rightarrow 0^{+}} \log |x| \\ \text { (c) } \lim _{x \rightarrow 0} \log |x| \end{array}$$

Evaluate each limit. (a) \(\lim _{x \rightarrow 0} \tan x\) (b) \(\lim _{x \rightarrow \infty / 2} \tan x\) (c) \(\lim _{x \rightarrow 3 \pi / 4} \tan x\)

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0}(x \ln |x|)\)

Find \(f^{\prime}(x)\) using the alternative definition. $$f(x)=-3 x^{2}+1$$

Evaluate each limit. (a) \(\lim _{x \rightarrow 4^{-}} \frac{x-4}{|x-4|}\) (b) \(\lim _{x \rightarrow 4^{+}} \frac{x-4}{|x-4|}\) (c) \(\lim _{x \rightarrow 4} \frac{x-4}{|x-4|}\)

Evaluate each limit. (a) \(\lim _{x \rightarrow 9} e^{\sqrt{x}}\) (b) \(\lim _{x \rightarrow-2} e^{\sqrt{x}}\) (c) \(\lim _{x \rightarrow 0} e^{\sqrt{x}}\)

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \tan \frac{1}{x}\)

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \cos \frac{1}{x}\)

Find \(f^{\prime}(x)\) using the alternative definition. $$f(x)=x^{2}+3 x-2$$

Write an expression for a function \(f(x)\) with the given features. \(f(x)\) is a quotient of two polynomials of degree greater than \(2, \lim _{x \rightarrow \infty} f(x)=0\)

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \cos \frac{1}{x}\)

Find \(f^{\prime}(x)\) using the alternative definition. $$f(x)=x^{3}$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \frac{2 x}{\tan x}\)

Find \(f^{\prime}(x)\) using the alternative definition. $$f(x)=1-x^{3}$$

Solve each problem. W(t)$ represents the gallons of water in a tank after I minutes. Complete the following. (a) Find the initial amount of water in the tank. (b) Find the amount of water in the rank after 12 minutes. (c) Is the rate of change in the amount of water in the rank constant? Explain. (d) Find the rate of change in the amount of water at 12 minutes. $$W(t)=500-t^{2}, \quad \text { for } 0 \leq t \leq 20$$

Find two functions \(f(x)\) and \(g(x)\) with the given properties. $$\lim _{x \rightarrow x} f(x)=\infty, \lim _{x \rightarrow x} g(x)=\infty, \text { and } \lim _{x \rightarrow x_{0}}[f(x)-g(x)]=\infty$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0}(x \csc x)\)

Solve each problem. W(t)$ represents the gallons of water in a tank after I minutes. Complete the following. (a) Find the initial amount of water in the tank. (b) Find the amount of water in the rank after 12 minutes. (c) Is the rate of change in the amount of water in the rank constant? Explain. (d) Find the rate of change in the amount of water at 12 minutes. $$W(t)=6000-20 t^{2}, \text { for } 0 \leq t \leq 15$$

Find two functions \(f(x)\) and \(g(x)\) with the given properties. $$\lim _{x \rightarrow \infty} f(x)=\infty, \lim _{x \rightarrow \infty} g(x)=\infty, \text { and } \lim _{x \rightarrow \infty}[f(x)-g(x)]=2$$

Use a table and/or graph to decide whether each limit exists. If a limit exists, find its value. \(\lim _{x \rightarrow 0} \frac{e^{x}-1}{2 x}\)

Solve each problem. W(t)$ represents the gallons of water in a tank after I minutes. Complete the following. (a) Find the initial amount of water in the tank. (b) Find the amount of water in the rank after 12 minutes. (c) Is the rate of change in the amount of water in the rank constant? Explain. (d) Find the rate of change in the amount of water at 12 minutes. $$W(t)=4 t+t^{2}, \text { for } 0 \leq t \leq 30$$

Find two functions \(f(x)\) and \(g(x)\) with the given properties. $$\lim _{x \rightarrow \infty} f(x)=0, \lim _{x \rightarrow \infty} g(x)=\infty, \text { and } \lim _{x \rightarrow \infty}[f(x) \cdot g(x)]=\infty$$

Solve each problem. W(t)$ represents the gallons of water in a tank after I minutes. Complete the following. (a) Find the initial amount of water in the tank. (b) Find the amount of water in the rank after 12 minutes. (c) Is the rate of change in the amount of water in the rank constant? Explain. (d) Find the rate of change in the amount of water at 12 minutes. $$W(t)=t^{3}+t+100, \text { for } 0 \leq t \leq 15$$

Find two functions \(f(x)\) and \(g(x)\) with the given properties. $$\lim _{x \rightarrow \infty} f(x)=0, \lim _{x \rightarrow \infty} g(x)=\infty, \text { and } \lim _{x \rightarrow \infty}[f(x) \cdot g(x)]=0$$

Assume that \(f(x)\) has domain \([0, \infty)\). Find \(\lim _{x \rightarrow x} f(x)\) if the graph of \(y=f(x)\) has oblique asymptote \(y=\frac{1}{2} x+3\)

Assume that \(f(x)\) has domain \([0, \infty)\). Find \(\lim f(x)\) if the graph of \(y=f(x)\) has oblique asymptote \(y=-2 x+3\)

Solve each problem. The weight of a small fish in grams after \(t\) weeks is modeled by $$W(t)=0.1 t^{2}$$. At what rate is the fish growing at time \(t=4 ?\)

Solve each problem.Use a calculator to answer each of the following. (a) From a graph of \(y=x e^{-x},\) what do you think is the value of \(\lim _{x \rightarrow \infty}\left(x e^{-x}\right) ?\) Support your answer by evaluating the function for several large values of \(x\). (b) Repeat part (a), but this time use the graph of the function \(y=x^{2} e^{-x}\) (c) On the basis of your results from parts (a) and (b), what do you think is the value of \(\lim \left(x^{n} e^{-x}\right)\) for other positive integers \(n ?\)

Solve each problem. A ball thrown straight up into the air has an initial height of 5 feet and an initial velocity of 128 feet per second. What is the velocity of the ball after 2 seconds?

Solve each problem. Use a calculator to answer each of the following. (a) From a graph of \(y=\frac{\ln x}{x},\) what do you think is the value of \(\lim _{x \rightarrow x} \frac{\ln x}{x} ?\) Support your answer by evaluating the function for several large values of \(x\). (b) Repeat part (a), but this time use the graph of the function \(y=\frac{(\ln x)^{2}}{x}\) (c) On the basis of your results from parts (a) and (b), what do you think is the value of \(\lim _{x \rightarrow \infty} \frac{(\ln x)^{x}}{x},\) where \(n\) is a positive integer?

Solve each problem. A helicopter is gradually rising straight up in the air. Its distance from the ground \(t\) seconds after takeoff is \(s(t)\) feet, where $$s(t)=t^{2}+t$$. (a) How long will it take for the helicopter to rise 20 feet? (b) Find the vertical velocity of the helicopter when it is 20 feet above the ground.

Solve each problem. Epidemiologists estimate that \(t\) days after the flu begins to spread in a small town, the percent of the population infected by the flu is approximated by $$p(t)=t^{2}+2 t$$ for \(0 \leq t \leq 5 .\) Find the instantaneous rate of change of the percent of the population infected at time \(t=3\).

MODELING Speed of a Skydiver If air resistance is not ignored during free fall, then the speed (in feet per second) of a skydiver after \(t\) seconds is given by $$ f(t)=176\left(1-e^{-a 2 t}\right) $$ Calculate \(\lim _{t \rightarrow x^{\infty}} f(t)\) and give an interpretation of its value. (IMAGES CANNOT COPY).

Solve each problem. The revenue (in thousands of dollars) from producing \(x\) units of an item is modeled by $$R(x)=10 x-0.002 x^{2}$$. Find the marginal revenue at \(x=1000\).

Solve each problem. Evans Price Adjustment Model If there is excess demand for a commodity, the price will rise rapidly at first and then more slowly, according to what economists call the "Evans price adjustment model." A typical function describing this behavior is given by $$ p(t)=12-4 e^{-0.5 t} $$ where \(p(t)\) is the price in dollars after \(t\) days. Calculate \(\lim _{t \rightarrow \infty} p(t)\) and give an interpretation of its value.

Solve each problem. Suppose that the total profit in hundreds of dollars from selling \(x\) items is given by $$P(x)=2 x^{2}-5 x+6$$. Find the marginal profit at \(x=2\).

Use a calculator to determine the derivative. A rumor is spreading through a city. The number of people who have heard the rumor after \(t\) days is modeled by $$f(t)=\frac{100,000}{1+9.134(0.8)^{t}}$$. Graph \(f(t)\) in the window \([0,32]\) by \([0,100,000] .\) Determine how fast the rumor is spreading after 8 days.

Recall from earlier work that the end behavior of the graph of a polynomial function is determined by the degree of the polynomial and the sign of the leading coefficient. Relate the concepts introduced earlier to those of this chapter. Rewrite the polynomial function $$ f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\dots+a_{1} x+a_{0} $$ as a product by factoring out the leading term, \(a_{n} x^{n}\)

Use a calculator to determine the derivative. When a drug is taken orally, the number of units of the drug in the bloodstream after \(t\) hours is modeled by the function $$f(t)=120\left(e^{-0.2 t}-e^{-t}\right).$$ Graph the function in the window \([0,16]\) by \([0,70]\). How many units of the drug are in the bloodstream after 6 hours? At what rate is the level of the drug in the blood. stream decreasing after 6 hours?

Recall from earlier work that the end behavior of the graph of a polynomial function is determined by the degree of the polynomial and the sign of the leading coefficient. Relate the concepts introduced earlier to those of this chapter. Determine \(\lim _{x \rightarrow x} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x)\) in each case. (a) \(a_{n}\) positive, \(n\) even (b) \(a_{n}\) negative, \(n\) even (c) \(a_{n}\) positive, \(n\) odd (d) \(a_{n}\) negative, \(n\) odd