Limits Cheat Sheet


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