Chapter 2

Einstein's Special Theory of Relativity says that the mass \(m(v)\) of an object is related to its velocity \(v\) by $$ m(v)=\frac{m_{0}}{\sqrt{1-v^{2} / c^{2}}} $$ Here \(m_{0}\) is the rest mass and \(c\) is the velocity of light. What is \(\lim _{v \rightarrow c^{-}} m(v)\) ?

Show that \(\cosh x\) is an even function.

Many software packages have programs for calculating limits, although you should be warned that they are not infallible. To develop confidence in your program, use it to recalculate some of the limits in Problems 1-28. Then for each of the following, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 2^{-}} \frac{x^{2}-x-2}{|x-2|} $$

A cab company charges \(\$ 2.50\) for the first \(\frac{1}{4}\) mile and \(\$ 0.20\) for each additional \(\frac{1}{8}\) mile. Sketch a graph of the cost of a cab ride as a function of the number of miles driven. Discuss the continuity of this function.

Prove that \(\lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{-n}=e\). Hint: First show that $$ \left(1-\frac{1}{n}\right)^{-n}=\left(1+\frac{1}{n-1}\right)^{n}=\left(1+\frac{1}{n-1}\right)^{n-1}\left(1+\frac{1}{n-1}\right) $$

Many software packages have programs for calculating limits, although you should be warned that they are not infallible. To develop confidence in your program, use it to recalculate some of the limits in Problems 1-28. Then for each of the following, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 1^{+}} \frac{2}{1+2^{1 /(x-1)}} $$

Use the Intermediate Value Theorem to prove that \(x^{3}+3 x-2=0\) has a real solution between 0 and \(1 .\)

Begin by plotting the function in an appropriate window. \(\lim _{x \rightarrow-\infty} \sqrt{\frac{2 x^{2}-3 x}{5 x^{2}+1}}\)

Since calculus software packages find \(\lim _{x \rightarrow a} f(x)\) by sampling a few values of \(f(x)\) for \(x\) near \(a\), they can be fooled. Find a function \(f\) for which \(\lim _{x \rightarrow 0} f(x)\) fails to exist but for which your software gives a value for the limit.

Use the Intermediate Value Theorem to prove that \((\cos t) t^{3}+6 \sin ^{5} t-3=0\) has a real solution between 0 and \(2 \pi\).

Begin by plotting the function in an appropriate window. \(\lim _{x \rightarrow-\infty}\left(\sqrt{2 x^{2}+3 x}-\sqrt{2 x^{2}-5}\right)\)

Use the Intermediate Value Theorem to show that \(x^{3}-7 x^{2}+14 x-8=0\) has at least one solution in the interval \([0,5]\). Sketch the graph of \(y=x^{3}-7 x^{2}+14 x-8\) over \([0,5]\). How many solutions does this equation really have?

Use the Intermediate Value Theorem to show that \(\sqrt{x}-\cos x=0\) has a solution between 0 and \(\pi / 2\). Zoom in on the graph of \(y=\sqrt{x}-\cos x\) to find an interval having length \(0.1\) that contains this solution.

Begin by plotting the function in an appropriate window. \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{10}\)

Show that the equation \(x^{5}+4 x^{3}-7 x+14=0\) has at least one real solution.

Begin by plotting the function in an appropriate window. \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}\)

Prove that \(f\) is continuous at \(c\) if and only if \(\lim _{t \rightarrow 0} f(c+t)=f(c)\).

Prove that if \(f\) is continuous at \(c\) and \(f(c)>0\) there is an interval \((c-\delta, c+\delta)\) such that \(f(x)>0\) on this interval.

Prove that if \(f\) is continuous on \([0,1]\) and satisfies \(0 \leq f(x) \leq 1\) there, then \(f\) has a fixed point; that is, there is a number \(c\) in \([0,1]\) such that \(f(c)=c\). Hint: Apply the Intermediate Value Theorem to \(g(x)=x-f(x)\).

Begin by plotting the function in an appropriate window. Your computer may indicate that some of these limits do not exist, but, if so, you should be able to interpret the answer as either \(\infty\) or \(-\infty\). \(\lim _{x \rightarrow 3^{-}} \frac{\sin |x-3|}{x-3}\)

Find the values of \(a\) and \(b\) so that the following function is continuous everywhere. $$ f(x)= \begin{cases}x+1 & \text { if } x<1 \\ a x+b & \text { if } 1 \leq x<2 \\\ 3 x & \text { if } x \geq 2\end{cases} $$

Begin by plotting the function in an appropriate window. Your computer may indicate that some of these limits do not exist, but, if so, you should be able to interpret the answer as either \(\infty\) or \(-\infty\). \(\lim _{x \rightarrow 3^{-}} \frac{\sin |x-3|}{\tan (x-3)}\)

Begin by plotting the function in an appropriate window. Your computer may indicate that some of these limits do not exist, but, if so, you should be able to interpret the answer as either \(\infty\) or \(-\infty\). \(\lim _{x \rightarrow 3^{-}} \frac{\cos (x-3)}{x-3}\)

Begin by plotting the function in an appropriate window. Your computer may indicate that some of these limits do not exist, but, if so, you should be able to interpret the answer as either \(\infty\) or \(-\infty\). \(\lim _{x \rightarrow \frac{\pi}{2}^{+}} \frac{\cos x}{x-\pi / 2}\)

Starting at 4 A.M., a hiker slowly climbed to the top of a mountain, arriving at noon. The next day, he returned along the same path, starting at \(5 \mathrm{~A} . \mathrm{M}\). and getting to the bottom at \(11 \mathrm{~A} . \mathrm{M}\). Show that at some point along the path his watch showed the same time on both days.

Begin by plotting the function in an appropriate window. Your computer may indicate that some of these limits do not exist, but, if so, you should be able to interpret the answer as either \(\infty\) or \(-\infty\). \(\lim _{x \rightarrow 0^{+}}(1+\sqrt{x})^{1 / \sqrt{x}}\)

The gravitational force exerted by the earth on an object having mass \(m\) that is a distance \(r\) from the center of the earth is $$ g(r)= \begin{cases}\frac{G M m r}{R^{3}}, & \text { if } r

Suppose that \(f\) is continuous on \([a, b]\) and it is never zero there. Is it possible that \(f\) changes sign on \([a, b]\) ? Explain.

Let \(f(x+y)=f(x)+f(y)\) for all \(x\) and \(y\) and suppose that \(f\) is continuous at \(x=0\). (a) Prove that \(f\) is continuous everywhere. (b) Prove that there is a constant \(m\) such that \(f(t)=m t\) for all \(t\) (see Problem 43 of Section 1.5).

Prove that if \(f(x)\) is a continuous function on an interval then so is the function \(|f(x)|=\sqrt{(f(x))^{2}}\).

Show that if \(g(x)=|f(x)|\) is continuous it is not necessarily true that \(f(x)\) is continuous.

Let \(f(x)=0\) if \(x\) is irrational and let \(f(x)=1 / q\) if \(x\) is the rational number \(p / q\) in reduced form \((q>0)\). (a) Sketch (as best you can) the graph of \(f\) on \((0,1)\). (b) Show that \(f\) is continuous at each irrational number in \((0,1)\), but is discontinuous at each rational number in \((0,1)\).