Chapter 2

Verify that the given equations are identities. \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\)

, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0}(\sin 5 x) / 3 x $$

Determine the largest interval over which the given function is continuous. $$ f(x)=\sec ^{-1} x, x \geq 0 $$

Give a rigorous proof that if \(\lim _{x \rightarrow \infty} f(x)=A\) and \(\lim _{x \rightarrow \infty} g(x)=B\), then $$ \lim _{x \rightarrow \infty}[f(x)+g(x)]=A+B $$

, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0} \cos (1 / x) $$

, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0} x \cos (1 / x) $$

A cell phone company charges \(\$ 0.12\) for connecting a call plus \(\$ 0.08\) per minute or any part thereof (e.g., a phone call lasting 2 minutes and 5 seconds costs \(\$ 0.12+3 \times \$ 0.08\) ). Sketch a graph of the cost of making a call as a function of the length of time \(t\) that the call lasts. Discuss the continuity of this function.

Find each of the following limits or indicate that it does not exist even in the infinite sense. (a) \(\lim _{x \rightarrow \infty} \sin x\) (b) \(\lim _{x \rightarrow \infty} \sin \frac{1}{x}\) (c) \(\lim _{x \rightarrow \infty} x \sin \frac{1}{x}\) (d) \(\lim _{x \rightarrow \infty} x^{3 / 2} \sin \frac{1}{x}\) (e) \(\lim _{x \rightarrow \infty} x^{-1 / 2} \sin x\) (f) \(\lim _{x \rightarrow \infty} \sin \left(\frac{\pi}{6}+\frac{1}{x}\right)\) (g) \(\lim _{x \rightarrow \infty} \sin \left(x+\frac{1}{x}\right)\) (h) \(\lim _{x \rightarrow \infty}\left[\sin \left(x+\frac{1}{x}\right)-\sin x\right]\)

, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 1} \frac{x^{3}-1}{\sqrt{2 x+2}-2} $$

A rental car company charges \(\$ 20\) for one day, allowing up to 200 miles. For each additional 100 miles, or any fraction thereof, the company charges \(\$ 18\). Sketch a graph of the cost for renting a car for one day as a function of the miles driven. Discuss the continuity of this function.

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 \(\lim _{v \rightarrow c^{-}} m(v) ?\)

, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0} \frac{x \sin 2 x}{\sin \left(x^{2}\right)} $$

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.

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

, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 2^{-}} \frac{x^{2}-x-2}{|x-2|} $$

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

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

, 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 \((\cos t) t^{3}+6 \sin ^{5} t-3=0\) 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 pauation really have?

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

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.

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

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. $$ f(x)=\left\\{\begin{array}{ll} x+1 & \text { if } x<1 \\ a x+b & \text { if } 1 \leq x<2 \\ 3 x & \text { if } x \geq 2 \end{array}\right. $$

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 A.M. and getting to the bottom at 11 A.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)=\left\\{\begin{array}{ll} \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 \(t\)

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

75\. 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 \((0,1)\).