Chapter Review

• 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)