Chapter 13

Find the following for the given function \(f:\) (a) \(f^{\prime}(a),\) where \(a\) is in the domain of \(f,\) and (b) \(f^{\prime}(3)\) and \(f^{\prime}(4)\) $$f(x)=x^{2}+2 x$$

Finding Limits Evaluate the limit if it exists. $$\lim _{x \rightarrow 7} \frac{\sqrt{x+2}-3}{x-7}$$

Limits of Sequences If the sequence with the given \(n\) th term is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{1}{3^{n}}$$

Estimating Limits Graphically Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 0} \frac{1}{1+e^{1 / x}}$$

Find the following for the given function \(f:\) (a) \(f^{\prime}(a),\) where \(a\) is in the domain of \(f,\) and (b) \(f^{\prime}(3)\) and \(f^{\prime}(4)\) $$f(x)=-\frac{1}{x^{2}}$$

Finding Limits Evaluate the limit if it exists. $$\lim _{h \rightarrow 0} \frac{\sqrt{1+h}-1}{h}$$

Limits of Sequences If the sequence with the given \(n\) th term is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{(-1)^{n}}{n}$$

One-Sided Limits Graph the piecewise-defined function and use your graph to find the values of the limits, if they exist. $$f(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } x \leq 2 \\ 6-x & \text { if } x>2 \end{array}\right.$$ (a) \(\lim _{x \rightarrow 2^{-}} f(x)\) (b) \(\lim _{x \rightarrow 2^{+}} f(x)\) (c) \(\lim _{x \rightarrow 2} f(x)\)

Find the following for the given function \(f:\) (a) \(f^{\prime}(a),\) where \(a\) is in the domain of \(f,\) and (b) \(f^{\prime}(3)\) and \(f^{\prime}(4)\) $$f(x)=\frac{x}{x+1}$$

Finding Limits Evaluate the limit if it exists. $$\lim _{x \rightarrow-4} \frac{\frac{1}{4}+\frac{1}{x}}{4+x}$$

Limits of Sequences If the sequence with the given \(n\) th term is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\sin (n \pi / 2)$$

One-Sided Limits Graph the piecewise-defined function and use your graph to find the values of the limits, if they exist. $$f(x)=\left\\{\begin{array}{ll} 2 & \text { if } x<0 \\ x+1 & \text { if } x \geq 0 \end{array}\right.$$ (a) \(\lim _{x \rightarrow 0^{-}} f(x)\) (b) \(\lim _{x \rightarrow 0^{+}} f(x)\) (c) \(\lim _{x \rightarrow 0} f(x)\)

Find the following for the given function \(f:\) (a) \(f^{\prime}(a),\) where \(a\) is in the domain of \(f,\) and (b) \(f^{\prime}(3)\) and \(f^{\prime}(4)\) $$f(x)=\sqrt{x-2}$$

Finding Limits Evaluate the limit if it exists. $$\lim _{t \rightarrow 0}\left(\frac{1}{t}-\frac{1}{t^{2}+t}\right)$$

Limits of Sequences If the sequence with the given \(n\) th term is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\cos n \pi$$

Tangent Lines (a) If \(f(x)=x^{3}-2 x+4,\) find \(f^{\prime}(a)\) (b) Find equations of the tangent lines to the graph of \(f\) at the points whose \(x\) -coordinates are \(0,1,\) and 2 (c) Graph \(f\) and the three tangent lines.

Finding Limits Evaluate the limit if it exists. $$\lim _{h \rightarrow 0} \frac{(3+h)^{-1}-3^{-1}}{h}$$

Limits of Sequences If the sequence with the given \(n\) th term is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{3}{n^{2}}\left[\frac{n(n+1)}{2}\right]$$

One-Sided Limits Graph the piecewise-defined function and use your graph to find the values of the limits, if they exist. $$f(x)=\left\\{\begin{array}{ll} 2 x+10 & \text { if } x \leq-2 \\ -x+4 & \text { if } x>-2 \end{array}\right.$$ (a) \(\lim _{x \rightarrow-2^{-}} f(x)\) (b) \(\lim _{x \rightarrow-2^{+}} f(x)\) (c) \(\lim _{x \rightarrow-2} f(x)\)

Tangent Lines (a) If \(g(x)=1 /(2 x-1),\) find \(g^{\prime}(a)\) (b) Find equations of the tangent lines to the graph of \(g\) at the points whose \(x\) -coordinates are \(-1,0,\) and 1 (c) Graph \(g\) and the three tangent lines.

Finding Limits Evaluate the limit if it exists. $$\lim _{t \rightarrow 4} \frac{\frac{1}{\sqrt{t}}-\frac{1}{2}}{t-4}$$

Limits of Sequences If the sequence with the given \(n\) th term is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{5}{n}\left(n+\frac{4}{n}\left[\frac{n(n+1)}{2}\right]\right)$$

A Function with Specified Limits Sketch the graph of an example of a function \(f\) that satisfies all of the following conditions. $$\begin{aligned} &\lim _{x \rightarrow 0^{-}} f(x)=2 \quad \lim _{x \rightarrow 0^{+}} f(x)=0\\\ &\lim _{x \rightarrow 2} f(x)=1 \quad f(0)=2 \quad f(2)=3 \end{aligned}$$ How many such functions are there?

The given limit represents the derivative of a function \(f\) at a number \(a\). Find \(f\) and \(a\) $$\lim _{h \rightarrow 0} \frac{(1+h)^{10}-1}{h}$$

Find the limit, and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 1} \frac{x^{2}-1}{\sqrt{x}-1}$$

Limits of Sequences If the sequence with the given \(n\) th term is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{24}{n^{3}}\left[\frac{n(n+1)(2 n+1)}{6}\right]$$

Graphing Calculator Pitfalls (a) Evaluate $$ h(x)=\frac{\tan x-x}{x^{3}} $$ for \(x=1,0.5,0.1,0.05,0.01,\) and 0.005 (b) Guess the value of \(\lim _{x \rightarrow 0} \frac{\tan x-x}{x^{3}}\) (c) Evaluate \(h(x)\) for successively smaller values of \(x\) until you finally get a value of 0 for \(h(x) .\) Are you still confident that your guess in part (b) is correct? Explain why you eventually got a value of 0 for \(h(x)\) (d) Graph the function \(h\) in the viewing rectangle \([-1,1]\) by \([0,1] .\) Then zoom in toward the point where the graph crosses the \(y\) -axis to estimate the limit of \(h(x)\) as \(x\) approaches \(0 .\) Continue to zoom in until you observe distortions in the graph of \(h .\) Compare with your results in part (c).

The given limit represents the derivative of a function \(f\) at a number \(a\). Find \(f\) and \(a\) $$\lim _{x \rightarrow 5} \frac{2^{x}-32}{x-5}$$

Find the limit, and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 0} \frac{(4+x)^{3}-64}{x}$$

Limits of Sequences If the sequence with the given \(n\) th term is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{12}{n^{4}}\left[\frac{n(n+1)}{2}\right]^{2}$$

The given limit represents the derivative of a function \(f\) at a number \(a\). Find \(f\) and \(a\) $$\lim _{t \rightarrow 1} \frac{\sqrt{t+1}-\sqrt{2}}{t-1}$$

Find the limit, and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow-1} \frac{x^{2}-x-2}{x^{3}-x}$$

A Function from a Description Find a formula from a function \(f\) that satisfies the following conditions. Vertical asymptotes \(x=1\) and \(x=3\) and horizontal asymptote \(y=1\)

The given limit represents the derivative of a function \(f\) at a number \(a\). Find \(f\) and \(a\) $$\lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h}$$

Find the limit, and use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 1} \frac{x^{8}-1}{x^{5}-1}$$

A Function from a Description Find a formula from a function \(f\) that satisfies the following conditions. $$\begin{aligned} &\lim _{x \rightarrow \infty} f(x)=0, \quad \lim _{x \rightarrow 0} f(x)=-\infty, \quad f(2)=0\\\ &\lim _{x \rightarrow 3^{-}} f(x)=\infty, \quad \lim _{x \rightarrow 3^{+}} f(x)=-\infty \end{aligned}$$

Velocity of a Ball If a ball is thrown straight up with a velocity of \(40 \mathrm{ft} / \mathrm{s}\), its height (in \(\mathrm{ft}\) ) after \(t\) seconds is given by \(y=40 t-16 t^{2} .\) Find the instantaneous velocity when \(t=2\)

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-4}|x+4|$$

A Function from a Description Find a formula from a function \(f\) that satisfies the following conditions. Asymptote Behavior How close to \(-3\) do we have to take \(x\) so that $$\frac{1}{(x+3)^{2}}>10,000$$

Velocity on the Moon If an arrow is shot upward on the moon with a velocity of \(58 \mathrm{m} / \mathrm{s},\) its height (in meters) after \(t\) seconds is given by \(H=58 t-0.83 t^{2}\) (a) Find the instantaneous velocity of the arrow after 1 second. (b) Find the instantaneous velocity of the arrow when \(t=a\) (c) At what time \(t\) will the arrow hit the moon? (d) With what velocity will the arrow hit the moon?

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-4^{-}} \frac{|x+4|}{x+4}$$

Show that $$\lim _{x \rightarrow \infty} f(x)=\lim _{t \rightarrow 0^{+}} f\left(\frac{1}{t}\right)$$ and $$\lim _{x \rightarrow-\infty} f(x)=\lim _{t \rightarrow 0^{-}} f\left(\frac{1}{t}\right)$$ if these limits exist.

Velocity of a Particle The displacement \(s\) (in meters) of a particle moving in a straight line is given by the equation of motion \(s=4 t^{3}+6 t+2,\) where \(t\) is measured in seconds. Find the instantancous velocity of the particle \(s\) at times \(t=a, t=1, t=2, t=3\)

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 2} \frac{|x-2|}{x-2}$$

(a) A tank contains \(5000 \mathrm{L}\) of pure water. Brine that contains \(30 \mathrm{g}\) of salt per liter of water is pumped into the tank at a rate of \(25 \mathrm{L} / \mathrm{min}\). Show that the con entration of salt after \(t\) minutes (in \(\mathrm{g} / \mathrm{L}\) ) is $$C(t)=\frac{30 t}{200+t}$$ (b) What happens to the concentration as \(t \rightarrow \infty ?\)

Inflating a Balloon A spherical balloon is being inflated. Find the rate of change of the surface area \(\left(S=4 \pi r^{2}\right)\) with respect to the radius \(r\) when \(r=2 \mathrm{ft}\).

The downward velocity of a falling raindrop at time \(t\) is modeled by the function $$v(t)=1.2\left(1-e^{-8.2 t}\right)$$ (a) Find the terminal velocity of the raindrop by evaluating \(\lim _{t \rightarrow \infty} v(t) .\) (Use the result of Example 3.) (b) Graph \(v(t),\) and use the graph to estimate how long it takes for the velocity of the raindrop to reach \(99 \%\) of its terminal velocity.

Does the Limit Exist? Find the limit, if it exists. If the limit does not exist, explain why. \(\lim _{x \rightarrow 1.5} \frac{2 x^{2}-3 x}{|2 x-3|}\)

(a) A sequence is defined recursively by \(a_{1}=0\) and $$a_{n+1}=\sqrt{2+a_{n}}$$ Find the first ten terms of this sequence rounded to eight decimal places. Does this sequence appear to be convergent? If so, guess the value of the limit. (b) Assuming that the sequence in part (a) is convergent, let \(\lim _{n \rightarrow \infty} a_{n}=L .\) Explain why \(\lim _{n \rightarrow \infty} a_{n+1}=L\) also and therefore $$L=\sqrt{2+L}$$ Solve this equation to find the exact value of \(L\)

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 0^{-}}\left(\frac{1}{x}-\frac{1}{|x|}\right)$$