Chapter 1

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 6} g(f(f(x))) $$

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 1} \frac{1}{x}=1 $$

Approximate the given limits both numerically and graphically. $$ \begin{array}{l} \lim _{x \rightarrow 2} f(x), \text { where } \\ f(x)=\left\\{\begin{array}{cl} x+2 & x \leq 2 \\ 3 x-5 & x>2 \end{array}\right. \end{array} $$

Evaluate the given limits using the graph of the function. \(f(x)=2^{x}+10\) (a) \(\lim _{x \rightarrow-\infty} f(x)\) (b) \(\lim _{x \rightarrow \infty} f(x)\)

Evaluate the given limits of the piecewise defined functions \(f\). $$ \begin{array}{ll} f(x)=\left\\{\begin{array}{cc} 2 x^{2}+5 x-1 & x<0 \\ \sin x & & x \geq 0 \end{array}\right. \\ \begin{array}{ll} \text { (a) } \lim _{x \rightarrow 0^{-}} f(x) & \text { (c) } \lim _{x \rightarrow 0} f(x) \\ \text { (b) } \lim _{x \rightarrow 0^{+}} f(x) & \text { (d) } f(0) \end{array} \end{array} $$

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 6} f(x) g(x)-f^{2}(x)+g^{2}(x) $$

Prove the given limit using an \(\varepsilon-\delta\) proof. \(\lim _{x \rightarrow 0} \sin x=0\) (Hint: use the fact that \(|\sin x| \leq|x|,\) with equality only when \(x=0 .)\)

Approximate the given limits both numerically and graphically. $$ \begin{array}{l} \lim _{x \rightarrow 3} f(x), \text { where } \\ f(x)=\left\\{\begin{array}{cl} x^{2}-x+1 & x \leq 3 \\ 2 x+1 & x>3 \end{array}\right. \end{array} $$

Numerically approximate the following limits: (a) \(\lim _{x \rightarrow 3^{-}} f(x)\) (b) \(\lim _{x \rightarrow 3^{+}} f(x)\) (c) \(\lim _{x \rightarrow 3} f(x)\) $$ f(x)=\frac{x^{2}-1}{x^{2}-x-6} $$

Evaluate the given limits of the piecewise defined functions \(f\). \(f(x)=\left\\{\begin{array}{cc}x^{2}-1 & x<-1 \\ x^{3}+1 & -1 \leq x \leq 1 \\\ x^{2}+1 & x>1\end{array}\right.\) (a) \(\lim _{x \rightarrow-1^{-}} f(x)\) (e) \(\lim _{x \rightarrow 1^{-}} f(x)\) (b) \(\lim _{x \rightarrow-1^{+}} f(x)\) (f) \(\lim _{x \rightarrow 1^{+}} f(x)\) (c) \(\lim _{x \rightarrow-1} f(x)\) (g) \(\lim _{x \rightarrow 1} f(x)\) (d) \(f(-1)\) (h) \(f(1)\)

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{l} \text { - } \lim _{x \rightarrow 1} f(x)=2, \quad \lim _{x \rightarrow 10} f(x)=1, \quad f(1)=1 / 5 \\ \text { - } \lim _{x \rightarrow 1} g(x)=0, \quad \lim _{x \rightarrow 10} g(x)=\pi, \quad g(10)=\pi \end{array} $$ $$ \lim _{x \rightarrow 1} f(x)^{g(x)} $$

Approximate the given limits both numerically and graphically. $$ \begin{array}{l} \lim _{x \rightarrow 0} f(x), \text { where } \\ f(x)=\left\\{\begin{array}{cc} \cos x & x \leq 0 \\ x^{2}+3 x+1 & x>0 \end{array}\right. \end{array} $$

Numerically approximate the following limits: (a) \(\lim _{x \rightarrow 3^{-}} f(x)\) (b) \(\lim _{x \rightarrow 3^{+}} f(x)\) (c) \(\lim _{x \rightarrow 3} f(x)\) $$ f(x)=\frac{x^{2}+5 x-36}{x^{3}-5 x^{2}+3 x+9} $$

Evaluate the given limits of the piecewise defined functions \(f\). $$ \begin{array}{ll} f(x)=\left\\{\begin{array}{ll} \cos x & x<\pi \\ \sin x & x \geq \pi \end{array}\right. & \\ \begin{array}{ll} \text { (a) } \lim _{x \rightarrow \pi^{-}} f(x) & \text { (c) } \lim _{x \rightarrow \pi} f(x) \\ \text { (b) } \lim _{x \rightarrow \pi^{+}} f(x) & & \text { (d) } f(\pi) \end{array} \end{array} $$

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{l} \text { - } \lim _{x \rightarrow 1} f(x)=2, \quad \lim _{x \rightarrow 10} f(x)=1, \quad f(1)=1 / 5 \\ \text { - } \lim _{x \rightarrow 1} g(x)=0, \quad \lim _{x \rightarrow 10} g(x)=\pi, \quad g(10)=\pi \end{array} $$ $$ \lim _{x \rightarrow 10} \cos (g(x)) $$

Approximate the given limits both numerically and graphically. $$ \begin{array}{l} \lim _{x \rightarrow \pi / 2} f(x), \text { where } \\ f(x)=\left\\{\begin{array}{ll} \sin x & x \leq \pi / 2 \\ \cos x & x>\pi / 2 \end{array}\right. \end{array} $$

Numerically approximate the following limits: (a) \(\lim _{x \rightarrow 3^{-}} f(x)\) (b) \(\lim _{x \rightarrow 3^{+}} f(x)\) (c) \(\lim _{x \rightarrow 3} f(x)\) $$ f(x)=\frac{x^{2}-11 x+30}{x^{3}-4 x^{2}-3 x+18} $$

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{l} \text { - } \lim _{x \rightarrow 1} f(x)=2, \quad \lim _{x \rightarrow 10} f(x)=1, \quad f(1)=1 / 5 \\ \text { - } \lim _{x \rightarrow 1} g(x)=0, \quad \lim _{x \rightarrow 10} g(x)=\pi, \quad g(10)=\pi \end{array} $$ $$ \lim _{x \rightarrow 1} f(x) g(x) $$

A function \(f\) and \(a\) value \(a\) are given. Approximate the limit of the difference quotient, \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h},\) using \(h=\pm 0.1, \pm 0.01 .\) $$ f(x)=-7 x+2, \quad a=3 $$

Numerically approximate the following limits: (a) \(\lim _{x \rightarrow 3^{-}} f(x)\) (b) \(\lim _{x \rightarrow 3^{+}} f(x)\) (c) \(\lim _{x \rightarrow 3} f(x)\) $$ f(x)=\frac{x^{2}-9 x+18}{x^{2}-x-6} $$

Evaluate the given limits of the piecewise defined functions \(f\). \(f(x)=\left\\{\begin{array}{cc}x+1 & x<1 \\ 1 & x=1 \\ x-1 & x>1\end{array}\right.\) (a) \(\lim _{x \rightarrow 1^{-}} f(x)\) (c) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow 1^{+}} f(x)\) (d) \(f(1)\)

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{l} \text { - } \lim _{x \rightarrow 1} f(x)=2, \quad \lim _{x \rightarrow 10} f(x)=1, \quad f(1)=1 / 5 \\ \text { - } \lim _{x \rightarrow 1} g(x)=0, \quad \lim _{x \rightarrow 10} g(x)=\pi, \quad g(10)=\pi \end{array} $$ $$ \lim _{x \rightarrow 1} g(5 f(x)) $$

A function \(f\) and \(a\) value \(a\) are given. Approximate the limit of the difference quotient, \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h},\) using \(h=\pm 0.1, \pm 0.01 .\) $$ f(x)=9 x+0.06, \quad a=-1 $$

Identify the horizontal and vertical asymptotes, if any, of the given function. $$ f(x)=\frac{2 x^{2}-2 x-4}{x^{2}+x-20} $$

Determine if \(f\) is continuous at the indicated values. If not, explain why. \(f(x)=\left\\{\begin{array}{cl}1 & x=0 \\ \frac{\sin x}{x} & x>0\end{array}\right.\) (a) \(x=0\) (b) \(x=\pi\)

Evaluate the given limits of the piecewise defined functions \(f\). \(\begin{array}{ll}f(x)=\left\\{\begin{array}{cl}x+1 & x<1 \\ 1 & x=1 \\ x-1 & x>1\end{array}\right. & \\ \begin{array}{rlr}\text { (a) } \lim _{x \rightarrow 1^{-}} f(x) & & \text { (c) } \lim _{x \rightarrow 1} f(x) \\\ \text { (b) } \lim _{x \rightarrow 1^{+}} f(x) & & \text { (d) } f(1)\end{array}\end{array}\)

Evaluate the given limit. $$ \lim _{x \rightarrow 3} x^{2}-3 x+7 $$

A function \(f\) and \(a\) value \(a\) are given. Approximate the limit of the difference quotient, \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h},\) using \(h=\pm 0.1, \pm 0.01 .\) $$ f(x)=x^{2}+3 x-7, \quad a=1 $$

Identify the horizontal and vertical asymptotes, if any, of the given function. $$ f(x)=\frac{-3 x^{2}-9 x-6}{5 x^{2}-10 x-15} $$

Determine if \(f\) is continuous at the indicated values. If not, explain why. \(f(x)=\left\\{\begin{array}{ll}x^{3}-x & x<1 \\ x-2 & x \geq 1\end{array}\right.\) (a) \(x=0\) (b) \(x=1\)

Evaluate the given limit. $$ \lim _{x \rightarrow \pi}\left(\frac{x-3}{x-5}\right)^{7} $$

A function \(f\) and \(a\) value \(a\) are given. Approximate the limit of the difference quotient, \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h},\) using \(h=\pm 0.1, \pm 0.01 .\) $$ f(x)=\frac{1}{x+1}, \quad a=2 $$

Identify the horizontal and vertical asymptotes, if any, of the given function. $$ f(x)=\frac{x^{2}+x-12}{7 x^{3}-14 x^{2}-21 x} $$

Determine if \(f\) is continuous at the indicated values. If not, explain why. \(f(x)=\left\\{\begin{array}{cl}\frac{x^{2}+5 x+4}{x^{2}+3 x+2} & x \neq-1 \\\ 3 & x=-1\end{array}\right.\) (a) \(x=-1\) (b) \(x=10\)

Evaluate the given limits of the piecewise defined functions \(f\). \(\begin{array}{ll}f(x)=\left\\{\begin{array}{cc}\frac{|x|}{x} & x \neq 0 \\ 0 & x=0\end{array}\right. & \\ \begin{array}{ll}\text { (a) } \lim _{x \rightarrow 0^{-}} f(x) & \text { (c) } \lim _{x \rightarrow 0} f(x) \\ \text { (b) } \lim _{x \rightarrow 0^{+}} f(x) & \text { (d) } f(0)\end{array}\end{array}\)

Evaluate the given limit. $$ \lim _{x \rightarrow \pi / 4} \cos x \sin x $$

A function \(f\) and \(a\) value \(a\) are given. Approximate the limit of the difference quotient, \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h},\) using \(h=\pm 0.1, \pm 0.01 .\) $$ f(x)=-4 x^{2}+5 x-1, \quad a=-3 $$

Identify the horizontal and vertical asymptotes, if any, of the given function. $$ f(x)=\frac{x^{2}-9}{9 x-9} $$

Determine if \(f\) is continuous at the indicated values. If not, explain why. \(f(x)=\left\\{\begin{array}{cl}\frac{x^{2}-64}{x^{2}-11 x+24} & x \neq 8 \\ 5 & x=8\end{array}\right.\) (a) \(x=0\) (b) \(x=8\)

Evaluate the limit: \(\lim _{x \rightarrow-1} \frac{x^{2}+5 x+4}{x^{2}-3 x-4}\).

Evaluate the given limit. $$ \lim _{x \rightarrow 1} \frac{2 x-2}{x+4} $$

A function \(f\) and \(a\) value \(a\) are given. Approximate the limit of the difference quotient, \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h},\) using \(h=\pm 0.1, \pm 0.01 .\) $$ f(x)=\ln x, \quad a=5 $$

Identify the horizontal and vertical asymptotes, if any, of the given function. $$ f(x)=\frac{x^{2}-9}{9 x+27} $$

Give the intervals on which the given function is continuous. $$ f(x)=x^{2}-3 x+9 $$

Evaluate the limit: \(\lim _{x \rightarrow-4} \frac{x^{2}-16}{x^{2}-4 x-32}\)

Evaluate the given limit. $$ \lim _{x \rightarrow 0} \ln x $$

A function \(f\) and \(a\) value \(a\) are given. Approximate the limit of the difference quotient, \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h},\) using \(h=\pm 0.1, \pm 0.01 .\) $$ f(x)=\sin x, \quad a=\pi $$

Identify the horizontal and vertical asymptotes, if any, of the given function. $$ f(x)=\frac{x^{2}-1}{-x^{2}-1} $$

Give the intervals on which the given function is continuous. $$ g(x)=\sqrt{x^{2}-4} $$

Evaluate the limit: \(\lim _{x \rightarrow-6} \frac{x^{2}-15 x+54}{x^{2}-6 x}\).