सीमाओं की सूची

यहाँ कुछ प्रमुख एवं महत्वपूर्ण गणितीय फलनों की सीमाएँ (limit) दी गई हैं। *a* और *b* दोनों नियतांक हैं (*x* के सापेक्ष)।

## सामान्य फलनों की सीमाएँ

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

::\lim_{x \to c} \, [f(x) \pm g(x)] = L_1 \pm L_2

::\lim_{x \to c} \, [f(x)g(x)] = L_1 \times 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} \, 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

:\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 (एल् हॉस्पिटल नियम L'Hôpital's rule)

:\lim_{h\to 0}{f(x+h)-f(x)\over h}=f'(x)

:\lim_{h\to0}\left(\frac{f(x+h)}{f(x)}\right)^\frac{1}{h}=\exp\left(\frac{f'(x)}{f(x)}\right)

:\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)

## उल्लेखनीय विशिष्ट सीमाएँ

: \lim_{x\to+\infty} \left(1+\frac{k}{x}\right)^{mx}=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+\text{...} +\sqrt{2}}}}}_n= \pi

:  \lim_{x \to 0} \left(\frac{a^x - 1}{x} \right) = \log{a}, \qquad \forall~a > 0

## सरल फलन

:\lim_{x \to c} a = a

:\lim_{x \to c} x = c

:\lim_{dux \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} 

## लघुगणकीय तथा चरघातांकी फलन
:\lim_{x\to1}\frac{\ln(x)}{x-1}=1

:\mbox{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

:\mbox{If } a 

::\lim_{x \to -\infty} a^x = \infty

## त्रिकोणमितीय फलन

:\lim_{x \to a} \sin x = \sin a

:\lim_{x \to a} \cos x = \cos a

यदि x रेडियन में हो तो:
:\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

:\lim_{x \to 0} \frac{\sin ax}{x} = a

:\lim_{x \to 0} \frac{\sin ax}{\sin bx} = \frac{a}{b}

## अनन्त के पास

:\lim_{x\to\infty}N/x=0 \text{ for any real }N 
:\lim_{x\to\infty}x/N=\begin{cases} \infty, & N > 0 \\ \text{does not exist}, & N = 0 \\ -\infty, & N 
:\lim_{x\to\infty}x^N=\begin{cases} \infty, & N > 0 \\ 1, & N = 0 \\ 0, & N 
:\lim_{x\to\infty}N^x=\begin{cases} \infty, & N > 1 \\ 1, & N = 1 \\ 0, & 0 
:\lim_{x\to\infty}N^{-x}=\lim_{x\to\infty}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 
:\lim_{x\to\infty}\sqrt[N]{x}= \infty \text{ for any } N > 0 
:\lim_{x\to\infty}\log x=\infty
:\lim_{x\to0^+}\log x=-\infty

श्रेणी:सीमा
श्रेणी:फलन