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{a}}[c\cdot{f(x)}]=c\cdot\lim_{x\to{a}}{f(x)}
\lim_{x\to{a}}[(f(x))^c]=(\lim_{x\to{a}}{f(x)})^c
\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{a}}[f(x)\cdot{g(x)}]=\lim_{x\to{a}}{f(x)}\cdot\lim_{x\to{a}}{g(x)}
\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


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 c}[f(x)g(x)]=\infty, \quad L>0
\lim_{x\to c}[f(x)g(x)]=-\infty, \quad L<0
\lim_{x\to c}\frac{g(x)}{f(x)}=0
\lim_{x\to \infty}(ax^n)=\infty, \quad a>0
\lim_{x\to -\infty}(ax^n)=\infty,\quad \mathrm{n \: is\: even} , \quad a>0
\lim_{x\to -\infty}(ax^n)=-\infty,\quad \mathrm{n \: is \: odd} , \quad a>0
\lim_{x\to \infty}\left(\frac{c}{x^a}\right)=0


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
Basic Limit \lim_{x\to{a}}{x}=a
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
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)})
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}
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
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