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
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যদি {\displaystyle \lim \_{x\to c}f(x)=L\_{1}} ও {\displaystyle \lim \_{x\to c}g(x)=L\_{2}} হয়, তবে

{\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 } যখন {\displaystyle L\_{2}\neq 0}
{\displaystyle \lim \_{x\to c}\,f(x)^{n}=L\_{1}^{n}\qquad } যখন n একটি ধনাত্মক সংখ্যা।
{\displaystyle \lim \_{x\to c}\,f(x)^{1 \over n}=L\_{1}^{1 \over n}\qquad } যখন n একটি ধনাত্মক সংখ্যা ও 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 } যখন {\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)}

কয়েকটি বিশেষ সীমা
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{\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 }

কয়েকটি সরল ফলাফলের সীমা
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{\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 } যখন r একটি 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 ও সূচকীয় ফলাফলের সীমা
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যদি {\displaystyle a>1\,} হয়, তবে

{\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, তবে

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

ত্রিকোণমিতীয় ফলাফলের সীমা
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{\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 } যেকোনো অখণ্ড সংখ্যা n র জন্য।

অসীমের দিকে
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{\displaystyle \lim \_{x\to \infty }N/x=0} 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}}}
{\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<N<1\end{cases}}}
{\displaystyle \lim \_{x\to \infty }N^{-x}=\lim \_{x\to \infty }1/N^{x}=0} যিকোনো {\displaystyle N>1} 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}}}
{\displaystyle \lim \_{x\to \infty }{\sqrt[{N}]{x}}=\infty } যেকোনো {\displaystyle N>0} র কারণে।
{\displaystyle \lim \_{x\to \infty }\log x=\infty }
{\displaystyle \lim \_{x\to 0^{+}}\log x=-\infty }

তথ্যসূত্র
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