Chapter 13

(a) Use a graph of $$f(x)=\frac{\sqrt{3+x}-\sqrt{3}}{x}$$ to estimate the value of \(\lim _{x \rightarrow 0} f(x)\) to two decimal places. (b) Use a table of values of \(f(x)\) to estimate the limit to four decimal places. (c) Use the Limit Laws to find the exact value of the limit.

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\frac{(-1)^{n}}{n}$$

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 one 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? (IMAGE CAN'T COPY).

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

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 instantaneous velocity of the particle \(s\) at times \(t=a\) \(t=1, t=2, t=3\)

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 limit, if it exists. If the limit does not exist, explain why. $$\text { 0. } \lim _{x \rightarrow-4^{-}} \frac{|x+4|}{x+4}$$

If the sequence is convergent, find its limit. If it is divergent, explain why. $$a_{n}=\cos n \pi$$

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 \(r=2 \mathrm{ft}\).

Graph the piecewise-defined function and use your graph to find the values of the limits, if they exist. $$f(x)=\left\\{\begin{array}{ll} -x+3 & \text { if } x<-1 \\ 3 & \text { if } x \geq-1 \end{array}\right.$$ (a) \(\lim _{x \rightarrow-1^{-}} f(x)\) (b) \(\lim _{x \rightarrow-1^{+}} f(x)\) (c) \(\lim _{x \rightarrow-1} f(x)\)

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

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

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) \(\lim _{x \rightarrow-2} f(x)\)

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 15} \frac{2 x^{2}-3 x}{|2 x-3|}$$

If the sequence 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 cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after \(t\) minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. $$\begin{array}{|c|c|}\hline t \text { (min) } & \text { Heartbeats } \\\\\hline 36 & 2530 \\\38 & 2661 \\\40 & 2806 \\\42 & 2948 \\\44 & 3080 \\\\\hline\end{array}$$ (a) Find the average heart rates (slopes of the secant lines over the time intervals \([40,42]\) and \([42,44]\) (b) Estimate the patient's heart rate after 42 minutes by averaging the slopes of these two secant lines.

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

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

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

A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume \(V\) of water remaining in the tank (in gallons) after \(t\) minutes. $$\begin{array}{|c|c|}\hline t \text { (min) } & V \text { (gal) } \\\\\hline 5 & 694 \\\10 & 444 \\\15 & 250 \\\20 &111 \\\25 & 28 \\\30 & 0 \\\\\hline\end{array}$$ (a) Find the average rates at which water flows from the tank (slopes of secant lines) for the time intervals \([10,15]\) and \([15,20]\) (b) The slope of the tangent line at the point \((15,250)\) represents the rate at which water is flowing from the tank after 15 minutes. Estimate this rate by averaging the slopes of the secant lines in part (a).

(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 reach 0 values for \(h(x) .\) Are you still confident that your guess in part (b) is correct? Explain why you eventually obtained 0 values. (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).

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

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

Let $$f(x)=\left\\{\begin{array}{ll}x-1 & \text { if } x<2 \\ x^{2}-4 x+6 & \text { if } x \geq 2\end{array}\right.$$ (a) Find \(\lim _{x \rightarrow 2}-f(x)\) and \(\lim _{x \rightarrow 2^{+}} f(x)\) (b) Does \(\lim _{x \rightarrow 2} f(x)\) exist? (c) Sketch the graph of \(f\)

(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 L/min. Show that the concentration of salt after \(t\) minutes (in grams per liter) is $$C(t)=\frac{30 t}{200+t}$$ (b) What happens to the concentration as \(t \rightarrow \infty ?\)

Let $$h(x)=\left\\{\begin{array}{ll}x & \text { if } x<0 \\ x^{2} & \text { if } 02\end{array}\right.$$ (a) Evaluate each limit if it exists. (i) \(\lim _{x \rightarrow 0^{+}} h(x)\) (iv) \(\lim _{x \rightarrow 2^{-}} h(x)\) (ii) \(\lim _{x \rightarrow 0} h(x)\) (v) \(\lim _{x \rightarrow 2^{+}} h(x)\) when (vi) \(\lim _{x \rightarrow 2} h(x)\) (b) Sketch the graph of \(h\)

Cancellation and Limits (a) What is wrong with the following equation? $$\frac{x^{2}+x-6}{x-2}=x+3$$ (b) In view of part (a), explain why the equation $$\lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x-2}=\lim _{x \rightarrow 2}(x+3)$$ is correct.

The Limit of a Recursive Sequence (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 \(L.\)

The Lorentz Contraction In the theory of relativity the Lorentz contraction formula $$L=L_{0} \sqrt{1-v^{2} / c^{2}}$$ expresses the length \(L\) of an object as a function of its velocity \(v\) with respect to an observer, where \(L_{0}\) is the length of the object at rest and \(c\) is the speed of light. Find \(\lim _{v \rightarrow c^{-}} L,\) and interpret the result. Why is a left-hand limit necessary?

Limits of Sums and Products (a) Show by means of an example that \(\lim _{x \rightarrow a}[f(x)+g(x)]\) may exist even though neither \(\lim _{x \rightarrow a} f(x)\) nor \(\lim _{x \rightarrow a} g(x)\) exists. (b) Show by means of an example that \(\lim _{x \rightarrow a}[f(x) g(x)]\) may exist even though neither \(\lim _{x \rightarrow a} f(x)\) nor \(\lim _{x \rightarrow a} g(x)\) exists.