Chapter 2

Find each of the right-hand and left-hand limits or state that they do not exist. $$\lim _{x \rightarrow 3^{+}} \frac{x-3}{\sqrt{x^{2}-9}}$$

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. \(f(x)=\frac{3}{x+1}\)

In Problems 41-52, verify that the given equations are identities. \(e^{-2 x}=\cosh 2 x-\sinh 2 x\)

Find each of the following limits or state that it does not exist. (a) \(\lim _{x \rightarrow 1^{+}} \sqrt{x-[x]}\) (b) \(\lim _{x \rightarrow 0^{+}}[1 / x]\) (c) \(\lim _{x \rightarrow 0^{+}} x(-1)^{[1 / x]}\) (d) \(\lim _{x \rightarrow 0^{+}}\left[x \rrbracket(-1)^{[1 / x]}\right.\)

Determine whether the function is continuous at the given point \(c\). If the function is not continuous, determine whether the discontinuity is removable or nonremovable. $$ f(x)=\frac{\cos x}{x} ; c=0 $$

Find each of the right-hand and left-hand limits or state that they do not exist. $$\lim _{x \rightarrow 1^{-}} \frac{\sqrt{1+x}}{4+4 x}$$

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. \(f(x)=\frac{3}{(x+1)^{2}}\)

In Problems 41-52, verify that the given equations are identities. \(\sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y\)

Determine whether the function is continuous at the given point \(c\). If the function is not continuous, determine whether the discontinuity is removable or nonremovable. $$ g(x)= \begin{cases}\frac{\sin x}{x}, & x \neq 0 \\ 0, & x=0\end{cases} $$

Find each of the right-hand and left-hand limits or state that they do not exist. $$\lim _{x \rightarrow 2^{+}} \frac{\left(x^{2}+1\right)[x]}{(3 x-1)^{2}}$$

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. \(F(x)=\frac{2 x}{x-3}\)

In Problems 41-52, verify that the given equations are identities. \(\sinh (x-y)=\sinh x \cosh y-\cosh x \sinh y\)

Find each of the following limits or state that it does not exist. (a) \(\lim _{x \rightarrow 3}[x] / x\) (b) \(\lim _{x \rightarrow 0^{+}} \llbracket x \rrbracket / x\) (c) \(\lim _{x \rightarrow 1.8}[x]\) (d) \(\lim _{x \rightarrow 1.8}[x] / x\)

Determine whether the function is continuous at the given point \(c\). If the function is not continuous, determine whether the discontinuity is removable or nonremovable. $$ F(x)=x \sin \frac{1}{x} ; c=0 $$

Find each of the right-hand and left-hand limits or state that they do not exist. $$\lim _{x \rightarrow 3^{-}}(x-[x])$$

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. \(F(x)=\frac{3}{9-x^{2}}\)

In Problems 41-52, verify that the given equations are identities. \(\cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y\)

CAS 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. $$ \lim _{x \rightarrow 0} \sqrt{x} $$

Determine whether the function is continuous at the given point \(c\). If the function is not continuous, determine whether the discontinuity is removable or nonremovable. $$ f(x)=\sin \frac{1}{x} ; c=0 $$

Find each of the right-hand and left-hand limits or state that they do not exist. $$\lim _{x \rightarrow 0^{-}} \frac{x}{|x|}$$

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. \(g(x)=\frac{14}{2 x^{2}+7}\)

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. $$ \lim _{x \rightarrow 0^{+}} x^{x} $$

Determine whether the function is continuous at the given point \(c\). If the function is not continuous, determine whether the discontinuity is removable or nonremovable. $$ f(x)=\frac{4-x}{2-\sqrt{x}} ; c=4 $$

Find each of the right-hand and left-hand limits or state that they do not exist. $$\lim _{x \rightarrow 3^{+}}\left[x^{2}+2 x\right]$$

Find the horizontal and vertical asymptotes for the graphs of the indicated functions. Then sketch their graphs. \(g(x)=\frac{2 x}{\sqrt{x^{2}+5}}\)

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. $$ \lim _{x \rightarrow 0} \sqrt{|x|} $$

In Problems 49-54, determine the largest interval over which the given function is continuous. $$ f(x)=\sqrt{25-x^{2}} $$

Suppose that \(f(x) g(x)=1\) for all \(x\) and \(\lim _{x \rightarrow a} g(x)=0\). Prove that \(\lim _{x \rightarrow a} f(x)\) does not exist.

The line \(y=a x+b\) is called an oblique asymptote to the graph of \(y=f(x)\) if either \(\lim _{x \rightarrow \infty}[f(x)-(a x+b)]=0\) or \(\lim _{x \rightarrow-\infty}[f(x)-(a x+b)]=0\). Find the oblique asymptote for $$ f(x)=\frac{2 x^{4}+3 x^{3}-2 x-4}{x^{3}-1} $$

In Problems 41-52, verify that the given equations are identities. \(\tanh (x-y)=\frac{\tanh x-\tanh y}{1-\tanh x \tanh y}\)

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. $$ \lim _{x \rightarrow 0}|x|^{x} $$

In Problems 49-54, determine the largest interval over which the given function is continuous. $$ f(x)=\frac{1}{\sqrt{25-x^{2}}} $$

Let \(R\) be the rectangle joining the midpoints of the sides of the quadrilateral \(Q\) having vertices \((\pm x, 0)\) and \((0, \pm 1)\). Calculate $$ \lim _{x \rightarrow 0^{+}} \frac{\text { perimeter of } R}{\text { perimeter of } Q} $$

Find the oblique asymptote for $$ f(x)=\frac{3 x^{3}+4 x^{2}-x+1}{x^{2}+1} $$

In Problems 41-52, verify that the given equations are identities. \(\sinh 2 x=2 \sinh x \cosh x\)

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. $$ \lim _{x \rightarrow 0}(\sin 2 x) / 4 x $$

In Problems 49-54, determine the largest interval over which the given function is continuous. $$ f(x)=\sin ^{-1} x $$

Using the symbols \(M\) and \(\delta\), give precise definitions of each expression. (a) \(\lim _{x \rightarrow c^{+}} f(x)=-\infty\) (b) \(\lim _{x \rightarrow c^{-}} f(x)=\infty\)

In Problems 41-52, verify that the given equations are identities. \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\)

In Problems 49-54, determine the largest interval over which the given function is continuous. $$ f(x)=\operatorname{sech} x $$

Using the symbols \(M\) and \(N\), give precise definitions of each expression. (a) \(\lim _{x \rightarrow \infty} f(x)=\infty\) (b) \(\lim _{x \rightarrow-\infty} f(x)=\infty\)

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. $$ \lim _{x \rightarrow 0} \cos (1 / x) $$

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

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. $$ \lim _{x \rightarrow 0} x \cos (1 / x) $$

In Problems 49-54, determine the largest interval over which the given function is continuous. $$ f(x)=\operatorname{sech}^{-1} x $$

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. $$ \lim _{x \rightarrow 1} \frac{x^{3}-1}{\sqrt{2 x+2}-2} $$

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) \(\lim _{x \rightarrow \infty}\left[\sin \left(x+\frac{1}{x}\right)-\sin x\right]\)

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. $$ \lim _{x \rightarrow 0} \frac{x \sin 2 x}{\sin \left(x^{2}\right)} $$

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.