# Limits Cheat Sheet

• Algebra
• Number Rules
• Expand Rules
• Fractions Rules
• Absolute Rules
• Exponent Rules
• Factor Rules
• Factorial Rules
• Log Rules
• Undefined
• Complex Number Rules
• Trigonometry
• Basic Identities
• Pythagorean Identities
• Double Angle Identities
• Sum Difference Identities
• Product To Sum Identities
• Triple Angle Identities
• Function Ranges
• Function Values
• Limits
• Limit Properties
• Limit to Infinity Properties
• Indeterminate Forms
• Common Limits
• Limit Rules
• Derivatives
• Derivatives Rules
• Common Derivatives
• Trigonometric Derivatives
• Arc Trigonometric Derivatives
• Hyperbolic Derivatives
• Arc Hyperbolic Derivatives
• Integrals
• Common Integrals
• Trigonometric Integrals
• Arc Trigonometric Integrals
• Hyperbolic Integrals
• Integrals of Special Functions
• Indefinite Integrals Rules
• Definite Integrals Rules

## Limit Properties

 \mathrm{If\:the\:limit\:of\:f(x),\:and\:g(x)\:exists,\:then\:the\:following\:apply:} \lim_{x\to a}(x}=a \lim \:_{x\to \:0}\left(x\right) \lim_{x\to{a}}[c\cdot{f(x)}]=c\cdot\lim_{x\to{a}}{f(x)} \lim \:_{x\to \:0}\left(5x\right) \lim_{x\to{a}}[(f(x))^c]=(\lim_{x\to{a}}{f(x)})^c \lim _{x\to 0}\left(\left(x\right)^{^7}\right) \lim_{x\to{a}}[f(x)\pm{g(x)}]=\lim_{x\to{a}}{f(x)}\pm\lim_{x\to{a}}{g(x)} \lim \:_{x\to \:9}\left(x+x^2\right) \lim_{x\to{a}}[f(x)\cdot{g(x)}]=\lim_{x\to{a}}{f(x)}\cdot\lim_{x\to{a}}{g(x)} \lim \:_{x\to \:4}\left(x^3x^2\right) \lim_{x\to{a}}[\frac{f(x)}{g(x)}]=\frac{\lim_{x\to{a}}{f(x)}}{\lim_{x\to{a}}{g(x)}}, \quad "where" \: \lim_{x\to{a}}g(x)\neq0 \lim _{x\to 3\:}\left(\frac{x^9}{x^7}\right)

## Limit to Infinity Properties

 \mathrm{For}\:\lim_{x\to c}f(x)=\infty, \lim_{x\to c}g(x)=L,\:\mathrm{the\:following\:apply:} \lim_{x\to c}[f(x)\pm g(x)]=\infty \:\lim _{x\to \infty }\left(\frac{1}{x}+x\right) \lim_{x\to c}[f(x)g(x)]=\infty, \quad L>0 \:\lim _{x\to \infty }\left(x^2\cdot 9\right) \lim_{x\to c}[f(x)g(x)]=-\infty, \quad L<0 \lim _{x\to \infty }\left(x^2\cdot \left(-7\right)\right) \lim_{x\to c}\frac{g(x)}{f(x)}=0 \lim _{x\to \infty}\left(\frac{x^{-1}}{x^2}\right) \lim_{x\to \infty}(ax^n)=\infty, \quad a>0 \:\lim _{x\to \infty }\left(7x^9\right) \lim_{x\to -\infty}(ax^n)=\infty,\quad \mathrm{n \: is\: even} , \quad a>0 \:\lim _{x\to -\infty }\left(15x^8\right) \lim_{x\to -\infty}(ax^n)=-\infty,\quad \mathrm{n \: is \: odd} , \quad a>0 \:\lim _{x\to -\infty }\left(15x^{19}\right) \lim_{x\to \infty}\left(\frac{c}{x^a}\right)=0 \:\lim _{x\to \infty }\left(\frac{15}{x^4}\right)

## Indeterminate Forms

 0^{0} \infty^{0} \frac{\infty}{\infty} \frac{0}{0} 0\cdot\infty \infty-\infty 1^{\infty}

## Common Limits

 \lim _{x\to \infty}((1+\frac{k}{x})^x)=e^k \lim _{x\to \infty}((\frac{x}{x+k})^x)=e^{-k} \lim _{x\to 0}((1+x)^{\frac{1}{x}})=e

## Limit Rules

 Limit of a constant \lim_{x\to{a}}{c}=c \lim _{x\to 0}5 Basic Limit \lim_{x\to{a}}{x}=a \lim _{x\to \:9}x Squeeze Theorem \mathrm{Let\:f,\:g\:and\:h\:be\:functions\:such\:that\:for\:all}\:x\in \left[a,\:b\right]\: \mathrm{(except\:possibly\:at\:the\:limit\:point\:c),\:} f\left(x\right)\le h\left(x\right)\le g\left(x\right) \mathrm{Also\:suppose\:that,\:}\lim _{x\to c}f\left(x\right)=\lim _{x\to c}g\left(x\right)=L \mathrm{Then\:for\:any\:}a\le c\le b,\:\lim _{x\to c}h\left(x\right)=L \lim _{x\to \infty \:\:}\left(\frac{\sin \left(x\right)}{x}\right) L'Hopital's Rule \mathrm{For}\:\lim_{x\to{a}}\left(\frac{f(x)}{g(x)}\right), \mathrm{if}\:\lim_{x\to{a}}\left(\frac{f(x)}{g(x)}\right)=\frac{0}{0}\:\mathrm{or}\:\lim_{x\to a}\left(\frac{f(x)}{g(x)}\right)=\frac{\pm\infty}{\pm\infty}, \mathrm{then}\quad\lim_{x\to{a}}(\frac{f(x)}{g(x)})=\lim_{x\to{a}}(\frac{f ^{'}(x)}{g ^{'}(x)}) \lim _{x\to \:\:0}\left(\frac{\sin \left(x\right)}{x}\right) Divergence Criterion \mathrm{If\:there\:exists\:two\:sequences,}\:\left\{x_n\right\}_{n=1}^{\infty }\mathrm{\:and\:}\left\{y_n\right\}_{n=1}^{\infty } \mathrm{with:} x_n\ne{c},\:\mathrm{and}\:y_n\ne{c} \lim{x_n}=\lim{y_n}=c \lim{f(x_n)}\ne\lim{f(y_n)} \mathrm{Then\:}\:\lim_{x\to\:c}f(x)\:\mathrm{does\:not\:exist} \lim _{x\to \infty \:\:}\left(\sin \left(x\right)\right) Limit Chain Rule \mathrm{if}\:\lim_{u \to b}f(u)=L,\:\mathrm{and}\:\lim_{x \to a}g(x)=b, \mathrm{and}\:f(x)\:\mathrm{is\:continuous\:at}\:x=b \mathrm{Then:}\:\lim_{x \to a} f(g(x))=L \lim _{x\to \frac{\pi \:}{2}}\left(\tan\left(3x\right)\right)
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