সীমা সর্বজনীন সম্পত্তি
সীমা (ইংরেজি: Limit) সম্বন্ধীয় কয়েকটি উপপাদ্য ও সাধারণত প্রচলিত কয়েকটি সীমা। এখানে a ও b হ'ল ধ্রুবক ও x হ'ল চলরাশি
কয়েকটি উপপাদ্য[সম্পাদনা]
যদি
ও
হয়, তবে
![{\displaystyle \lim _{x\to c}\,[f(x)\pm g(x)]=L_{1}\pm L_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb00a9174f995bae626da947886d0229d14b275)
![{\displaystyle \lim _{x\to c}\,[f(x)g(x)]=L_{1}\times L_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d8693d558a7dfc0f5f0c5900a05dd950f6f7f1)
যখন ![{\displaystyle L_{2}\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/367d7118c2a411867307c1b631ea41b4a19c9887)
যখন n একটি ধনাত্মক সংখ্যা।
যখন n একটি ধনাত্মক সংখ্যা ও যদি n যুগ্ম সংখ্যা, ![{\displaystyle L_{1}>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35473d31115e840d8e588eee571a235d7c0583bc)
যখন
বা
(L'Hôpital's rule)
![{\displaystyle \lim _{h\to 0}{f(x+h)-f(x) \over h}=f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6148d980c98953dbd86c98078c4d653bedc41af)
![{\displaystyle \lim _{h\to 0}\left({\frac {f(x+h)}{f(x)}}\right)^{\frac {1}{h}}=\exp \left({\frac {f'(x)}{f(x)}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a11d7e228e328f7c206e0bdf7a26f2b8684c7d4c)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/942c956bb9d9415e620bf752f4e37d9b8cbaaf4e)
কয়েকটি বিশেষ সীমা[সম্পাদনা]
![{\displaystyle \lim _{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{mx}=e^{mk}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/add4edf8e48c9e29210a0df91c0c28a4c6fb3a80)
![{\displaystyle \lim _{x\to +\infty }\left(1-{\frac {1}{x}}\right)^{x}={\frac {1}{e}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9db2ba69ebe50f31110495ba9c7d0c45e2754281)
![{\displaystyle \lim _{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{x}=e^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d33901e97ce95f7ea5f47ffdd47d0c30b1ec47d3)
![{\displaystyle \lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}=e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e67d9f7e2588c9b3d418f1107e9ea27b8f330ed)
![{\displaystyle \lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+{\text{...}}+{\sqrt {2}}}}}}}} _{n}=\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/15782054b42f9fbb4be35ecb85a1de97abfc6709)
কয়েকটি সরল ফলাফলের সীমা[সম্পাদনা]
![{\displaystyle \lim _{x\to c}a=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4aaff459f69192f8dc759314582c6751a41363)
![{\displaystyle \lim _{x\to c}x=c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/337e2c532e2cedd5b0d67bd903d39cd1cbacaf77)
![{\displaystyle \lim _{x\to c}ax+b=ac+b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d30f7a105c48fca95512fc7752493eb7dfa6d29c)
যখন r একটি ধনাত্মক সংখ্যা
![{\displaystyle \lim _{x\to 0^{+}}{\frac {1}{x^{r}}}=+\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/97a337010a50e6173a4b1c1b5953ac7ed9ed4bab)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67a6eb1d5a7ff61ca5cc3d44d6ba1b8171bce2d9)
ঘাতাংকীয় ও সূচকীয় ফলাফলের সীমা[সম্পাদনা]
যদি
হয়, তবে
![{\displaystyle \lim _{x\to 0^{+}}\log _{a}x=-\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/72808181c55d9096253749e5968f71c47c87bd9b)
![{\displaystyle \lim _{x\to \infty }\log _{a}x=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bac08a36cc877e299461440d1c9c08d88373f041)
![{\displaystyle \lim _{x\to -\infty }a^{x}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5300504623b781a3067f7f6a88dfb5bf0ffb46f)
যদি
হয়, তবে
![{\displaystyle \lim _{x\to -\infty }a^{x}=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f573f1a9b7b255c1116ee111dfc915dcbc61a113)
ত্রিকোণমিতীয় ফলাফলের সীমা[সম্পাদনা]
![{\displaystyle \lim _{x\to a}\sin x=\sin a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04bf4a31eeee313433f14413d9e3c441d69576a9)
![{\displaystyle \lim _{x\to a}\cos x=\cos a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/233fccfb3bbfdad201662ecc5dce951fd4baf7b9)
![{\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f2fb52f5211c7b7aa69d9e75195afaab5b9d5b1)
![{\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e68c22c5c70aa65f792c06b2cf2fb6e1ecfe16b)
![{\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x^{2}}}={\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/580636aba444eda89b876cfc9e36e1f31d40e771)
যেকোনো অখণ্ড সংখ্যা n র জন্য।
যেকোনো বাস্তব সংখ্যা N র জন্য।
![{\displaystyle \lim _{x\to \infty }x/N={\begin{cases}\infty ,&N>0\\{\text{does not exist}},&N=0\\-\infty ,&N<0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bfb0df72e56cbb50ee3aa8bf08c9448aac14aa9)
![{\displaystyle \lim _{x\to \infty }x^{N}={\begin{cases}\infty ,&N>0\\1,&N=0\\0,&N<0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d65f9d24a6d8cf90b0d955281e7ad9c6f78a182a)
![{\displaystyle \lim _{x\to \infty }N^{x}={\begin{cases}\infty ,&N>1\\1,&N=1\\0,&0<N<1\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4095d481ad66b14be21345ba0bcfb1cc9c102e)
যিকোনো
র কারণে।
![{\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{N}}={\begin{cases}1,&N>0\\0,&N=0\\{\text{does not exist}},&N<0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61204dd0899445181689dd595b467af963c3150e)
যেকোনো
র কারণে।
![{\displaystyle \lim _{x\to \infty }\log x=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a3dc39adf241c3465eb63b5e002296b7d0c57e)
![{\displaystyle \lim _{x\to 0^{+}}\log x=-\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7a454bbcf3fafcf6ea82fdcd1b8346c5c0d1a7)