Chapter 2

Find \(\lim _{x \rightarrow 2}[f(x)-f(2)] /(x-2)\) for each given function \(f\). $$f(x)=3 x^{2}+2 x+1$$

Find the limits. \(\lim _{\theta \rightarrow \pi^{+}} \frac{\theta^{2}}{\sin \theta}\)

Use \(\log _{a} x=(\ln x) /(\ln a)\) to calculate each of the logarithms in Problems 33-36. \(\log _{5} 12\)

Sketch the graph of $$ f(x)=\left\\{\begin{aligned} -x & \text { if } x<0 \\ x & \text { if } 0 \leq x<1 \\ 1+x & \text { if } x \geq 1 \end{aligned}\right. $$ Then find each of the following or state that it does not exist. (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow 1} f(x)\) (c) \(f(1)\) (d) \(\lim _{x \rightarrow 1^{+}} f(x)\)

In Problems 24-35, at what points, if any, are the functions discontinuous? $$ g(x)= \begin{cases}x^{2} & \text { if } x<0 \\ -x & \text { if } 0 \leq x \leq 1 \\ x & \text { if } x>1\end{cases} $$

Find \(\lim _{x \rightarrow 2}[f(x)-f(2)] /(x-2)\) for each given function \(f\). $$f(x)=\frac{1}{x}$$

Find the limits. \(\lim _{x \rightarrow 3^{-}} \frac{x^{3}}{x-3}\)

Use \(\log _{a} x=(\ln x) /(\ln a)\) to calculate each of the logarithms in Problems 33-36. \(\log _{7}(0.11)\)

Sketch the graph of $$ g(x)=\left\\{\begin{aligned} -x+1 & \text { if } x<1 \\ x-1 & \text { if } 1

In Problems 24-35, at what points, if any, are the functions discontinuous? $$ f(t)=\llbracket t \rrbracket $$

Find \(\lim _{x \rightarrow 2}[f(x)-f(2)] /(x-2)\) for each given function \(f\). $$f(x)=\frac{3}{x^{2}}$$

Find the limits. \(\lim _{\theta \rightarrow(\pi / 2)^{+}} \frac{\pi \theta}{\cos \theta}\)

Use \(\log _{a} x=(\ln x) /(\ln a)\) to calculate each of the logarithms in Problems 33-36. \(\log _{11}(8.12)^{1 / 5}\)

Sketch the graph of \(f(x)=x-[x]\); then find each of the following or state that it does not exist. (a) \(f(0)\) (b) \(\lim _{x \rightarrow 0} f(x)\) (c) \(\lim _{x \rightarrow 0^{-}} f(x)\) (d) \(\lim _{x \rightarrow 1 / 2} f(x)\)

Find the limits. \(\lim _{x \rightarrow 3^{-}} \frac{x^{2}-x-6}{x-3}\)

Use \(\log _{a} x=(\ln x) /(\ln a)\) to calculate each of the logarithms in Problems 33-36. \(\log _{10}(8.57)^{7}\)

Sketch the graph of a function \(f\) that satisfies all the following conditions. (a) Its domain is \([-2,2]\). (b) \(f(-2)=f(-1)=f(1)=f(2)=1\). (c) It is discontinuous at \(-1\) and 1 . (d) It is right continuous at \(-1\) and left continuous at 1 .

Find the limits. \(\lim _{x \rightarrow 2^{+}} \frac{x^{2}+2 x-8}{x^{2}-4}\)

In Problems 37-40, use natural logarithms to solve each of the exponential equations. Hint: To solve \(3^{x}=11\), take ln of both sides, obtaining \(x \ln 3=\ln 11\); then \(x=(\ln 11) /(\ln 3) \approx 2.1827\). \(2^{x}=17\)

Find \(\lim _{x \rightarrow 1}\left(x^{2}-1\right) /|x-1|\) or state that it does not exist.

Sketch the graph of a function that has domain \([0,2]\) and is continuous on \([0,2)\) but not on \([0,2]\).

Prove that \(\lim _{x \rightarrow c} f(x)=L \Leftrightarrow \lim _{x \rightarrow c}[f(x)-L]=0\).

Find the limits. \(\lim _{x \rightarrow 0^{+}} \frac{[x]}{x}\)

In Problems 37-40, use natural logarithms to solve each of the exponential equations. Hint: To solve \(3^{x}=11\), take ln of both sides, obtaining \(x \ln 3=\ln 11\); then \(x=(\ln 11) /(\ln 3) \approx 2.1827\). \(5^{x}=13\)

Evaluate \(\lim _{x \rightarrow 0}(\sqrt{x+2}-\sqrt{2}) / x\). Hint: Rationalize the numerator by multiplying the numerator and denominator by \(\sqrt{x+2}+\sqrt{2}\)

Sketch the graph of a function that has domain \([0,6]\) and is continuous on \([0,2]\) and \((2,6]\) but is not continuous on \([0,6]\).

Prove that \(\lim _{x \rightarrow c} f(x)=0 \Leftrightarrow \lim _{x \rightarrow c}|f(x)|=0\).

Find the limits. \(\lim _{x \rightarrow 0^{-}} \frac{[x]}{x}\)

In Problems 37-40, use natural logarithms to solve each of the exponential equations. Hint: To solve \(3^{x}=11\), take ln of both sides, obtaining \(x \ln 3=\ln 11\); then \(x=(\ln 11) /(\ln 3) \approx 2.1827\). \(5^{-2 x-3}=4\)

Let \(f(x)=\left\\{\begin{aligned} x & \text { if } x \text { is rational } \\\\-x & \text { if } x \text { is irrational } \end{aligned}\right.\) Find each value, if possible. (b) \(\lim _{x \rightarrow 0} f(x)\) (a) \(\lim _{x \rightarrow 1} f(x)\)

Sketch the graph of a function that has domain \([0,6]\) and is continuous on \((0,6)\) but not on \([0,6]\).

Prove that \(\lim _{x \rightarrow c}|x|=|c|\).

Find the limits. \(\lim _{x \rightarrow 0^{-}} \frac{|x|}{x}\)

In Problems 37-40, use natural logarithms to solve each of the exponential equations. Hint: To solve \(3^{x}=11\), take ln of both sides, obtaining \(x \ln 3=\ln 11\); then \(x=(\ln 11) /(\ln 3) \approx 2.1827\). \(12^{1 /(6-1)}=4\)

Sketch, as best you can, the graph of a function \(f\) that satisfies all the following conditions. (a) Its domain is the interval \([0,4]\). (b) \(f(0)=f(1)=f(2)=f(3)=f(4)=1\) (c) \(\lim _{x \rightarrow 1} f(x)=2\) (d) \(\lim _{x \rightarrow 2} f(x)=1\) (e) \(\lim _{x \rightarrow 3^{-}} f(x)=2\) (f) \(\lim _{x \rightarrow 3^{+}} f(x)=1\)

Let $$ f(x)=\left\\{\begin{aligned} x & \text { if } x \text { is rational } \\ -x & \text { if } x \text { is irrational } \end{aligned}\right. $$ Sketch the graph of this function as best you can and decide where it is continuous.

Find examples to show that if (a) \(\lim _{x \rightarrow c}[f(x)+g(x)]\) exists, this does not imply that either \(\lim _{x \rightarrow c} f(x)\) or \(\lim _{x \rightarrow c} g(x)\) exists; (b) \(\lim _{x \rightarrow c}[f(x) \cdot g(x)]\) exists, this does not imply that either \(\lim _{x \rightarrow c} f(x)\) or \(\lim _{x \rightarrow c} g(x)\) exists.

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

Let \(f(x)= \begin{cases}x^{2} & \text { if } x \text { is rational } \\ x^{4} & \text { if } x \text { is irrational }\end{cases}\) For what values of \(a\) does \(\lim _{x \rightarrow a} f(x)\) exist?

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. $$ f(x)=\sin x ; c=0 $$

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

Find the limits. \(\lim _{x \rightarrow 0^{-}} \frac{1+\cos x}{\sin x}\)

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

The function \(f(x)=x^{2}\) had been carefully graphed, but during the night a mysterious visitor changed the values of \(f\) at a million different places. Does this affect the value of \(\lim _{x \rightarrow a} f(x)\) at any \(a\) ? Explain.

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. $$ f(x)=\frac{x^{2}-100}{x-10} ; c=10 $$

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

Find the limits. \(\lim _{x \rightarrow \infty} \frac{\sin x}{x}\)

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

Find each of the following limits or state that it does not exist. (a) \(\lim _{x \rightarrow 1} \frac{|x-1|}{x-1}\) (b) \(\lim _{x \rightarrow 1^{-}} \frac{|x-1|}{x-1}\) (c) \(\lim _{x \rightarrow 1^{-}} \frac{x^{2}-|x-1|-1}{|x-1|}\) (d) \(\lim _{x \rightarrow 1^{-}}\left[\frac{1}{x-1}-\frac{1}{|x-1|}\right]\)

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. $$ f(x)=\frac{\sin x}{x} ; c=0 $$