Calculus Course/Limit

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Contents

List of LimitEdit

This is a list of limits for common functions. Note that a and b are constants with respect to x.

Limits for general functionsEdit

If \lim_{x\to c}f(x)=L_1 and \lim_{x\to c}g(x)=L_2 then:

\lim_{x\to c}\Big[f(x)\pm g(x)\Big]=L_1\pm L_2
\lim_{x\to c}\Big[f(x)\cdot g(x)\Big]=L_1\cdot L_2
\lim_{x\to c}\frac{f(x)}{g(x)}=\frac{L_1}{L_2}\qquad\text{ if } L_2\ne 0
\lim_{x\to c}f(x)^n=L_1^n\qquad\text{ if }n\text{ is a positive integer}
\lim_{x \to c}\sqrt[n]{f(x)}=\sqrt[n]{L_1}\qquad\text{ if }n\text{ is a positive integer, and if }n\text{ is even, then }L_1>0
\lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f'(x)}{g'(x)}\qquad\text{ if }\lim_{x\to c}f(x)=\lim_{x\to c}g(x)=0\text{ or }\lim_{x\to c}\bigl|g(x)\bigr|=\infty (L'Hôpital's rule)

Limits of general functionsEdit

\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=f'(x)
\lim_{h\to 0}\left(\frac{f(x+h)}{f(x)}\right)^\frac{1}{h}=\exp\left(\frac{f'(x)}{f(x)}\right)
\lim_{h\to 0}\left(\frac{f(x(1+h))}{f(x)}\right)^\frac{1}{h}=\exp\left(\frac{x f'(x)}{f(x)}\right)

Notable special limitsEdit

\lim_{x\to\infty}\left(1+\frac{k}{x}\right)^m=e^{mk}
\lim_{x\to\infty}\left(1-\frac{1}{x}\right)^x=\frac{1}{e}
\lim_{x\to\infty}\left(1+\frac{k}{x}\right)^x=e^k
\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}=e
\lim_{n\to\infty}2^n\underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\cdot+\sqrt2}}}}_n=\pi

Simple functionsEdit

\lim_{x\to c}a=a
\lim_{x\to c}x=c
\lim_{x\to c}ax+b=ac+b
\lim_{x\to c}x^r=c^r\qquad\mbox{ if }r\mbox{ is a positive integer}
\lim_{x\to 0^+}\frac{1}{x^r}=\infty
\lim_{x\to 0^-}\frac{1}{x^r}=\begin{cases} -\infty, & \text{if } r \text{ is odd} \\ \infty, & \text{if } r \text{ is even}\end{cases}

Logarithmic and exponential functionsEdit

For a>1 :

\lim_{x\to 0^+}\log_a(x)=-\infty
\lim_{x\to\infty}\log_a(x)=\infty
\lim_{x\to-\infty}a^x=0
\lim_{x\to\infty}a^x=\infty

Trigonometric functionsEdit

\lim_{x\to a}\sin(x)=\sin(a)
\lim_{x\to a}\cos(x)=\cos(a)
\lim_{x\to 0}\frac{\sin(x)}{x}=1
\lim_{x\to 0}\frac{1-\cos(x)}{x}=0
\lim_{x\to 0}\frac{1-\cos(x)}{x^2}=\frac{1}{2}
\lim_{x\to n^\pm}\tan\left(\pi x+\frac{\pi}{2}\right)=\mp\infty\qquad\text{for any integer }n

Near infinitiesEdit

\lim_{x\to\infty}\frac{N}{x}=0 \text{ for any real }N
\lim_{x\to\infty}\frac{x}{N}=\begin{cases}\infty, & N>0 \\ \text{does not exist}, & N=0 \\ -\infty, & N<0\end{cases}
\lim_{x\to\infty}x^N=\begin{cases}\infty, & N>0 \\ 1, & N=0 \\ 0, & N<0\end{cases}
\lim_{x\to\infty}N^x=\begin{cases}\infty, & N>1 \\ 1, & N=1 \\ 0, & -1<N<1 \end{cases}
\lim_{x\to\infty}N^{-x}=\lim_{x\to\infty}\frac{1}{N^x}=0\text{ for any }N>1
\lim_{x\to\infty}\sqrt[x]{N}=\begin{cases} 1, & N>0 \\ 0, & N=0 \\ \text{does not exist}, & N<0 \end{cases}
\lim_{x\to\infty}\sqrt[N]{x}=\infty\text{ for any }N>0
\lim_{x\to\infty}\log(x)=\infty
\lim_{x\to 0^+}\log(x)=-\infty

ReferenceEdit