Chapter Review

Printed Page 139

• Slope of a secant line: $m_{\sec }= \dfrac{f( x) -f( c) }{x-c}$ (p. 70)
• Slope of a tangent line $m_{\tan }=\lim\limits_{x\rightarrow c} \dfrac{f( x) -f( c) }{x-c}$ (p. 70)
• The limit $L$ of a function $y=f(x)$ as $x$ approaches a number $c$ does not depend on the value of $f$ at $c$. (p. 74)
• The limit $L$ of a function $y=f( x)$ as $x$ approaches a number $c$ is unique. A function cannot have more than one limit as $x$ approaches $c$. (p. 74)
• The limit $L$ of a function $y=f( x)$ as $x$ approaches a number $c$ exists if and only if both one-sided limits exist at $c$ and both one-sided limits are equal. That is, $\lim\limits_{x\rightarrow c}f( x) =L$ if and only if $\lim\limits_{x\rightarrow c^{-}}f( x) =\lim\limits_{x\rightarrow c^{+}}f( x) =L$. (p. 74)

Basic Limits

• $\lim\limits_{x\rightarrow c} A=A$, $A$ a constant (p. 81)
• $\lim\limits_{x\rightarrow c}x=c$ (p. 81)

Properties of Limits If $f$ and $g$ are functions for which $\lim\limits_{x\rightarrow c}f( x)$ and $\lim\limits_{x\rightarrow c}g(x)$ both exist and if $k$ is any real number, then:

• $\lim\limits_{x\rightarrow c}[f(x)\pm g(x)]=\lim\limits_{x\rightarrow c}f(x)\pm \lim\limits_{x\rightarrow c}g(x)$ (p. 82)
• $\lim\limits_{x\rightarrow c}[f(x)\cdot g(x)]=\lim\limits_{x\rightarrow c}f(x)\cdot \lim\limits_{x\rightarrow c}g(x)$ (p. 82)
• $\lim\limits_{x\rightarrow c}[kg(x)]=k\lim\limits_{x\rightarrow c}\,g(x)$ (p. 83)
• $\lim\limits_{x\rightarrow c}[f(x)]^{n}=\left[ \lim\limits_{\kern.5ptx\rightarrow c}f(x)\right] ^{n}$, $n\geq 2$ is an integer (p. 84)
• $\lim\limits_{x\rightarrow c}\sqrt[n]{f(x)}=\sqrt[n]{ \lim\limits_{x\rightarrow c}f(x)}$, provided $f(x)\ge 0$ if $n$ is even (p. 85)
• $\lim\limits_{x\rightarrow c}[f(x)]^{m/n}=\left[ \lim\limits_{\kern.5ptx\rightarrow c}f(x)\right] ^{m/n}$, provided $[f( x)] ^{m/n}$ is defined for positive integers $m$ and $n$ (p. 85)
• $\lim\limits_{x\rightarrow c}\left[ \dfrac{f(x)}{g(x)}\right] = \dfrac{\lim\limits_{x\rightarrow c}f(x)}{\lim\limits_{x\rightarrow c}g(x) }$, provided $\lim\limits_{x\rightarrow c}g(x)\neq 0$ (p. 87)
• If $P$ is a polynomial function, then $\lim\limits_{x \rightarrow c}P(x)=P(c)$. (p. 86)
• If $R$ is a rational function and if $c$ is in the domain of $R$, then $\lim\limits_{x\rightarrow c}R( x) =R( c)$. (p. 87)

Definitions

• Continuity at a number (p. 93)
• Removable discontinuity (p. 95)
• One-sided continuity at a number (p. 95)
• Continuity on an interval (p. 96)
• Continuity on a domain (p. 97)

Properties of Continuity

• A polynomial function is continuous on its domain. (p. 97)
• A rational function is continuous on its domain. (p. 97)
• If the functions $f$ and $g$ are continuous at a number $c$, and if $k$ is a real number, then the functions $f+g$, $f-g$, $f\cdot g$ and $kf$ are also continuous at $c$. If $g(c)$ ≠ 0, the function $\dfrac{f}{g}$ is continuous at $c.$ (p. 98)
• If a function $g$ is continuous at $c$ and a function $f$ is continuous at $g( c)$, then the composite function $(f\circ g)(x)=f(g(x))$ is continuous at $c$. (p. 99)
• If $f$ is a one-to-one function that is continuous on its domain, then its inverse function $f^{-1}$ is also continuous on its domain. (p. 100)

The Intermediate Value Theorem Let $f$ be a function that is continuous on a closed interval $[a,b]$ with $f(a)\neq f(b)$. If $N$ is any number between $f(a)$ and $f(b)$, then there is at least one number $c$ in the open interval $(a,b)$ for which $f(c)=N$. (p. 100)

Basic Limits

• $\lim\limits_{\theta\rightarrow 0}\dfrac{\sin \theta }{\theta }=1$ (p. 108)
• $\lim\limits_{x\rightarrow c}\sin x=\sin c$ (p. 111)
• $\lim\limits_{x\rightarrow c}\cos x=\cos c$ (p. 111)
• $\lim\limits_{\theta \rightarrow 0}\dfrac{\cos \theta -1}{\theta }=0$ (p. 111)
• $\lim\limits_{x\rightarrow c}a^{x}=a^{c}$; $a>0$,  $a\neq 1$ (p. 114)
• $\lim\limits_{x\rightarrow c}\log _{a}x=\log _{a}c$; $a>0,$ $a\neq 1$, and $c>0$ (p. 114)

Squeeze Theorem If the functions $f$, $g$, and $h$ have the property that for all $x$ in an open interval containing $c$, except possibly at $c$, $f(x)\leq g(x)\leq h(x)$, and if $\lim\limits_{x\rightarrow c}f(x)=\lim\limits_{x\rightarrow c}h(x)=L$, then $\lim\limits_{x\rightarrow c}g(x)=L.$ (p. 107)

Properties of Continuity

• The six trigonometric functions are continuous on their domains. (pp. 111-113)
• The six inverse trigonometric functions are continuous on their domains. (p. 113)
• An exponential function is continuous on its domain. (p. 114)
• A logarithmic function is continuous on its domain. (p. 114)

Basic Limits

• $\lim\limits_{x\rightarrow 0^{-}}\dfrac{1}{x}=-\infty;$ $\lim\limits_{x\rightarrow 0^{+}}\dfrac{1}{x}=\infty$ (p. 118)
• $\lim\limits_{x\rightarrow \infty }\dfrac{1}{x}=0$; $\lim\limits_{x\rightarrow -\infty }\dfrac{1}{x}=0$ (p. 120)
• $\lim\limits_{x\rightarrow 0}\dfrac{1}{x^{2}}=\infty$ (p. 117)
• $\lim\limits_{x\rightarrow 0^{+}}\ln x=-\infty$ (p. 118)
• $\lim\limits_{x\rightarrow \infty}\ln x=\infty$ (p. 124)
• $\lim\limits_{x\rightarrow -\infty }e^{x}=0;$ $\lim\limits_{x\rightarrow \infty }e^{x}=\infty$ (p. 124)

Definitions

• Vertical asymptote (p. 119)
• Horizontal asymptote (p. 125)

Properties of Limits at Infinity (p. 120): If $k$ is a real number, $n\geq 2$ is an integer, and the functions $f$ and $g$ approach real numbers as $x\rightarrow \infty$, then:

• $\lim\limits_{x\rightarrow \infty } A=A$, where $A$ is a number
• $\lim\limits_{x\rightarrow \infty } [kf(x)]=k\lim\limits_{x\rightarrow \infty }f(x)$
• $\lim\limits_{x\rightarrow \infty }[f(x)\pm g(x)]=\lim\limits_{x\rightarrow \infty }f(x)\pm \lim\limits_{x\rightarrow \infty }g(x)$
• $\lim\limits_{x\rightarrow \infty }[f(x)g(x)]=\left[ \lim\limits_{\kern.5ptx\rightarrow \infty }f(x)\right] \left[\lim\limits_{x\rightarrow \infty }g(x)\right]$
• $\lim\limits_{x\rightarrow \infty }\dfrac{f(x)}{g(x)}=\dfrac{\lim\limits_{x\rightarrow \infty }f(x)}{\lim\limits_{x\rightarrow \infty}g(x)}$ if $\lim\limits_{x\rightarrow \infty }g(x)\neq 0$
• $\lim\limits_{x\rightarrow \infty }[f(x)]^{n}=\left[\lim\limits_{x\rightarrow \infty }f(x)\right] ^{n}$
• $\lim\limits_{x\rightarrow \infty }\sqrt[n]{f(x)}=\sqrt[n]{\lim\limits_{x\rightarrow \infty }f(x)}$, where $f(x)\geq 0$ if $n$ is even

Definitions

• Limit of a Function (p. 132)
• Limit at Infinity (p. 136)
• Infinite Limit (p. 137)
• Infinite Limit at Infinity (p. 137)

Properties of limits

• If $\lim\limits_{x\rightarrow c}f(x)\gt0$, then there is an open interval around $c$, for which $f(x)\gt0$ everywhere in the interval, except possibly at $c$. (p. 136)
• If $\lim\limits_{x\rightarrow c}f(x)\lt0$, then there is an open interval around $c$, for which $f(x)\lt0$ everywhere in the interval, except possibly at $c$. (p. 136)