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| {{unreferenced|date=September 2012}}
| | Marvella is what you can call her but it's not the most female name out there. One of the very best things in the globe for him is to collect badges but he is having difficulties to find time for it. Managing people is his occupation. Puerto Rico is where he and his wife live.<br><br>Feel free to visit my web page; at home std test ([http://hsmlibrary.dothome.co.kr/xe/board_koQG15/407574 click through the up coming document]) |
| {{Expand list|date=February 2011}}
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| This is a '''list of [[limit (mathematics)|limit]]s''' for common [[function (mathematics)|function]]s. Note that ''a'' and ''b'' are constants with respect to ''x''.
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| ==Limits for general functions==
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| :<math>\text{If }\lim_{x \to c} f(x) = L_1 \text{ and }\lim_{x \to c} g(x) = L_2 \text{ then:}</math>
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| ::<math>\lim_{x \to c} \, [f(x) \pm g(x)] = L_1 \pm L_2</math>
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| ::<math>\lim_{x \to c} \, [f(x)g(x)] = L_1 \times L_2</math>
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| ::<math>\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L_1}{L_2} \qquad \text{ if } L_2 \ne 0</math>
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| ::<math>\lim_{x \to c} \, f(x)^n = L_1^n \qquad \text{ if }n \text{ is a positive integer}</math>
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| ::<math>\lim_{x \to c} \, f(x)^{1 \over n} = L_1^{1 \over n} \qquad \text{ if }n \text{ is a positive integer, and if } n \text{ is even, then } L_1 > 0</math>
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| :<math>\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} g(x) = \pm\infty</math> ([[L'Hôpital's rule]])
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| ==Limits of general functions==
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| :<math>\lim_{h\to 0}{f(x+h)-f(x)\over h}=f'(x)</math>
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| :<math>\lim_{h\to0}\left(\frac{f(x+h)}{f(x)}\right)^\frac{1}{h}=\exp\left(\frac{f'(x)}{f(x)}\right)</math>
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| :<math>\lim_{h \to 0}{ \left({f(x(1+h))\over{f(x)}}\right)^{1\over{h}} }=\exp\left(\frac{x f'(x)}{f(x)}\right)</math>
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| ==Notable special limits==
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| : <math>\lim_{x\to+\infty} \left(1+\frac{k}{x}\right)^{mx}=e^{mk}</math> | |
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| : <math>\lim_{x\to+\infty} \left(1-\frac{1}{x}\right)^x=\frac{1}{e}</math>
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| : <math>\lim_{x\to+\infty} \left(1+\frac{k}{x}\right)^x=e^k</math>
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| : <math>\lim_{n\to\infty} \frac{n}{\sqrt[n]{n!}}=e</math>
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| : <math>\lim_{n\to \infty }\, 2^{n} \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\text{...} +\sqrt{2}}}}}_n= \pi</math>
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| : <math> \lim_{x \to 0} \left( \frac{a^x - 1}{x} \right) = \ln{a}, \qquad \forall~a > 0</math>
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| ==Simple functions==
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| :<math>\lim_{x \to c} a = a</math>
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| :<math>\lim_{x \to c} x = c</math>
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| :<math>\lim_{x \to c} ax + b = ac + b</math>
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| :<math>\lim_{x \to c} x^r = c^r \qquad \mbox{ if } r \mbox{ is a positive integer}</math>
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| :<math>\lim_{x \to 0^+} \frac{1}{x^r} = +\infty</math>
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| :<math>\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} </math>
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| ==Logarithmic and exponential functions==
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| :<math>\lim_{x\to1}\frac{\ln(x)}{x-1}=1</math>
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| :<math>\mbox{For } a > 1: \,</math>
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| ::<math>\lim_{x \to 0^+} \log_a x = -\infty</math>
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| ::<math>\lim_{x \to \infty} \log_a x = \infty</math>
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| ::<math>\lim_{x \to -\infty} a^x = 0</math>
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| :<math>\mbox{If } a < 1: \,</math>
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| ::<math>\lim_{x \to -\infty} a^x = \infty</math>
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| ==Trigonometric functions==
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| :<math>\lim_{x \to a} \sin x = \sin a</math>
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| :<math>\lim_{x \to a} \cos x = \cos a</math>
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| If <math>x</math> is expressed in radians:
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| :<math>\lim_{x \to 0} \frac{\sin x}{x} = 1</math>
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| :<math>\lim_{x \to 0} \frac{1-\cos x}{x} = 0</math>
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| :<math>\lim_{x \to 0} \frac{1-\cos x}{x^2} = \frac{1}{2}</math>
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| :<math>\lim_{x \to n^\pm} \tan \left(\pi x + \frac{\pi}{2}\right) = \mp\infty \qquad \text{for any integer } n</math>
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| :<math>\lim_{x \to 0} \frac{\sin ax}{x} = a</math>
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| :<math>\lim_{x \to 0} \frac{\sin ax}{\sin bx} = \frac{a}{b}</math>
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| ==Near infinities==
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| :<math>\lim_{x\to\infty}N/x=0 \text{ for any real }N </math>
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| :<math>\lim_{x\to\infty}x/N=\begin{cases} \infty, & N > 0 \\ \text{does not exist}, & N = 0 \\ -\infty, & N < 0 \end{cases}</math>
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| :<math>\lim_{x\to\infty}x^N=\begin{cases} \infty, & N > 0 \\ 1, & N = 0 \\ 0, & N < 0 \end{cases}</math>
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| :<math>\lim_{x\to\infty}N^x=\begin{cases} \infty, & N > 1 \\ 1, & N = 1 \\ 0, & 0 < N < 1 \end{cases}</math>
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| :<math>\lim_{x\to\infty}N^{-x}=\lim_{x\to\infty}1/N^{x}=0 \text{ for any } N > 1</math>
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| :<math>\lim_{x\to\infty}\sqrt[x]{N}=\begin{cases} 1, & N > 0 \\ 0, & N = 0 \\ \text{does not exist}, & N < 0 \end{cases}</math>
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| :<math>\lim_{x\to\infty}\sqrt[N]{x}= \infty \text{ for any } N > 0 </math>
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| :<math>\lim_{x\to\infty}\log x=\infty</math>
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| :<math>\lim_{x\to0^+}\log x=-\infty</math>
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| [[Category:Limits (mathematics)]]
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| [[Category:Mathematics-related lists|Limits]]
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| [[Category:Functions and mappings]]
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Marvella is what you can call her but it's not the most female name out there. One of the very best things in the globe for him is to collect badges but he is having difficulties to find time for it. Managing people is his occupation. Puerto Rico is where he and his wife live.
Feel free to visit my web page; at home std test (click through the up coming document)