সমললৈ যাওক

# সীমাৰ তালিকা

সীমা (ইংৰাজী: Limit) সম্বন্ধীয় কেইটামান উপপাদ্য আৰু সাধাৰণতে প্ৰচলিত কেইটামান সীমা। ইয়াত a আৰু b হ'ল ধ্ৰুৱক আৰু x হ'ল চলক।

## কেইটামান উপপাদ্য

যদি ${\displaystyle \lim _{x\to c}f(x)=L_{1}}$ আৰু ${\displaystyle \lim _{x\to c}g(x)=L_{2}}$ তেন্তে

${\displaystyle \lim _{x\to c}\,[f(x)\pm g(x)]=L_{1}\pm L_{2}}$
${\displaystyle \lim _{x\to c}\,[f(x)g(x)]=L_{1}\times L_{2}}$
${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {L_{1}}{L_{2}}}\qquad }$ যদিহে ${\displaystyle L_{2}\neq 0}$
${\displaystyle \lim _{x\to c}\,f(x)^{n}=L_{1}^{n}\qquad }$ যদিহে n এটা ধনাত্মক সংখ্যা।
${\displaystyle \lim _{x\to c}\,f(x)^{1 \over n}=L_{1}^{1 \over n}\qquad }$ যদিহে n এটা ধনাত্মক সংখ্যা আৰু যদি n যুগ্ম সংখ্যা, ${\displaystyle L_{1}>0}$
${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}\qquad }$ যদিহে ${\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0}$ বা ${\displaystyle \lim _{x\to c}|g(x)|=+\infty }$ (L'Hôpital's rule)
${\displaystyle \lim _{h\to 0}{f(x+h)-f(x) \over h}=f'(x)}$
${\displaystyle \lim _{h\to 0}\left({\frac {f(x+h)}{f(x)}}\right)^{\frac {1}{h}}=\exp \left({\frac {f'(x)}{f(x)}}\right)}$
${\displaystyle \lim _{h\to 0}{\left({f(x(1+h)) \over {f(x)}}\right)^{1 \over {h}}}=\exp \left({\frac {xf'(x)}{f(x)}}\right)}$

## কেইটামান বিশেষ সীমা

${\displaystyle \lim _{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{mx}=e^{mk}}$
${\displaystyle \lim _{x\to +\infty }\left(1-{\frac {1}{x}}\right)^{x}={\frac {1}{e}}}$
${\displaystyle \lim _{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{x}=e^{k}}$
${\displaystyle \lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}=e}$
${\displaystyle \lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+{\text{...}}+{\sqrt {2}}}}}}}} _{n}=\pi }$

## কেইটামান সৰল ফলনৰ সীমা

${\displaystyle \lim _{x\to c}a=a}$
${\displaystyle \lim _{x\to c}x=c}$
${\displaystyle \lim _{x\to c}ax+b=ac+b}$
${\displaystyle \lim _{x\to c}x^{r}=c^{r}\qquad }$ যদিহে r এটা ধনাত্মক সংখ্যা
${\displaystyle \lim _{x\to 0^{+}}{\frac {1}{x^{r}}}=+\infty }$
${\displaystyle \lim _{x\to 0^{-}}{\frac {1}{x^{r}}}={\begin{cases}-\infty ,&{\text{if }}r{\text{ is odd}}\\+\infty ,&{\text{if }}r{\text{ is even}}\end{cases}}}$

## ঘাতাংকীয় আৰু সূচকীয় ফলনৰ সীমা

যদি ${\displaystyle a>1\,}$ হয়, তেন্তে

${\displaystyle \lim _{x\to 0^{+}}\log _{a}x=-\infty }$
${\displaystyle \lim _{x\to \infty }\log _{a}x=\infty }$
${\displaystyle \lim _{x\to -\infty }a^{x}=0}$

যদি ${\displaystyle a<1\,}$ হয়, তেন্তে

${\displaystyle \lim _{x\to -\infty }a^{x}=\infty }$

## ত্ৰিকোণমিতীয় ফলনৰ সীমা

${\displaystyle \lim _{x\to a}\sin x=\sin a}$
${\displaystyle \lim _{x\to a}\cos x=\cos a}$
${\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}$
${\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}=0}$
${\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x^{2}}}={\frac {1}{2}}}$
${\displaystyle \lim _{x\to n^{\pm }}\tan \left(\pi x+{\frac {\pi }{2}}\right)=\mp \infty \qquad }$ যিকোনো অখণ্ড সংখ্যা n ৰ বাবে।

## অসীমৰ সমীপত

${\displaystyle \lim _{x\to \infty }N/x=0}$ যিকোনো বাস্তৱ সংখ্যা N ৰ বাবে।
${\displaystyle \lim _{x\to \infty }x/N={\begin{cases}\infty ,&N>0\\{\text{does not exist}},&N=0\\-\infty ,&N<0\end{cases}}}$
${\displaystyle \lim _{x\to \infty }x^{N}={\begin{cases}\infty ,&N>0\\1,&N=0\\0,&N<0\end{cases}}}$
${\displaystyle \lim _{x\to \infty }N^{x}={\begin{cases}\infty ,&N>1\\1,&N=1\\0,&0
${\displaystyle \lim _{x\to \infty }N^{-x}=\lim _{x\to \infty }1/N^{x}=0}$ যিকোনো ${\displaystyle N>1}$ ৰ কাৰণে।
${\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{N}}={\begin{cases}1,&N>0\\0,&N=0\\{\text{does not exist}},&N<0\end{cases}}}$
${\displaystyle \lim _{x\to \infty }{\sqrt[{N}]{x}}=\infty }$ যিকোনো ${\displaystyle N>0}$ ৰ কাৰণে।
${\displaystyle \lim _{x\to \infty }\log x=\infty }$
${\displaystyle \lim _{x\to 0^{+}}\log x=-\infty }$