সীমার তালিকা

সীমা (ইংরেজি: Limit) সম্বন্ধীয় কয়েকটি উপপাদ্য ও সাধারণত প্রচলিত কয়েকটি সীমা। এখানে a ও b হ'ল ধ্রুবক ও x হ'ল চলরাশি যদি lim x → c f ( x ) = L 1 {\displaystyle \lim _{x\to c}f(x)=L_{1}} ও lim x → c g ( x ) = L 2 {\displaystyle \lim _{x\to c}g(x)=L_{2}} হয়, তবে lim x → c [ f ( x ) ± g ( x ) ] = L 1 ± L 2 {\displaystyle \lim _{x\to c}\,[f(x)\pm g(x)]=L_{1}\pm L_{2}} lim x → c [ f ( x ) g ( x ) ] = L 1 × L 2 {\displaystyle \lim _{x\to c}\,[f(x)g(x)]=L_{1}\times L_{2}} lim x → c f ( x ) g ( x ) = L 1 L 2 {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {L_{1}}{L_{2}}}\qquad } যখন L 2 ≠ 0 {\displaystyle L_{2}\neq 0} lim x → c f ( x ) n = L 1 n {\displaystyle \lim _{x\to c}\,f(x)^{n}=L_{1}^{n}\qquad } যখন n একটি ধনাত্মক সংখ্যা। lim x → c f ( x ) 1 n = L 1 1 n {\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, L 1 > 0 {\displaystyle L_{1}>0} lim x → c f ( x ) g ( x ) = lim x → c f ′ ( x ) g ′ ( x ) {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}\qquad } যখন lim x → c f ( x ) = lim x → c g ( x ) = 0 {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0} বা lim x → c | g ( x ) | = + ∞ {\displaystyle \lim _{x\to c}|g(x)|=+\infty } (L'Hôpital's rule) lim h → 0 f ( x + h ) − f ( x ) h = f ′ ( x ) {\displaystyle \lim _{h\to 0}{f(x+h)-f(x) \over h}=f'(x)} lim h → 0 ( f ( x + h ) f ( x ) ) 1 h = exp ⁡ ( f ′ ( x ) 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)} lim h → 0 ( f ( x ( 1 + h ) ) f ( x ) ) 1 h = exp ⁡ ( x f ′ ( x ) f ( x ) ) {\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)} lim x → + ∞ ( 1 + k x ) m x = e m k {\displaystyle \lim _{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{mx}=e^{mk}} lim x → + ∞ ( 1 − 1 x ) x = 1 e {\displaystyle \lim _{x\to +\infty }\left(1-{\frac {1}{x}}\right)^{x}={\frac {1}{e}}} lim x → + ∞ ( 1 + k x ) x = e k {\displaystyle \lim _{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{x}=e^{k}} lim n → ∞ n n ! n = e {\displaystyle \lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}=e} lim n → ∞ 2 n 2 − 2 + 2 + ... + 2 ⏟ n = π {\displaystyle \lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+{\text{...}}+{\sqrt {2}}}}}}}} _{n}=\pi } lim x → c a = a {\displaystyle \lim _{x\to c}a=a} lim x → c x = c {\displaystyle \lim _{x\to c}x=c} lim x → c a x + b = a c + b {\displaystyle \lim _{x\to c}ax+b=ac+b} lim x → c x r = c r {\displaystyle \lim _{x\to c}x^{r}=c^{r}\qquad } যখন r একটি ধনাত্মক সংখ্যা lim x → 0 + 1 x r = + ∞ {\displaystyle \lim _{x\to 0^{+}}{\frac {1}{x^{r}}}=+\infty } lim x → 0 − 1 x r = { − ∞ , if  r  is odd + ∞ , if  r  is even {\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}}} যদি a > 1 {\displaystyle a>1\,} হয়, তবে lim x → 0 + log a ⁡ x = − ∞ {\displaystyle \lim _{x\to 0^{+}}\log _{a}x=-\infty } lim x → ∞ log a ⁡ x = ∞ {\displaystyle \lim _{x\to \infty }\log _{a}x=\infty } lim x → − ∞ a x = 0 {\displaystyle \lim _{x\to -\infty }a^{x}=0} যদি a < 1 {\displaystyle a<1\,} হয়, তবে lim x → − ∞ a x = ∞ {\displaystyle \lim _{x\to -\infty }a^{x}=\infty } lim x → a sin ⁡ x = sin ⁡ a {\displaystyle \lim _{x\to a}\sin x=\sin a} lim x → a cos ⁡ x = cos ⁡ a {\displaystyle \lim _{x\to a}\cos x=\cos a} lim x → 0 sin ⁡ x x = 1 {\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1} lim x → 0 1 − cos ⁡ x x = 0 {\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}=0} lim x → 0 1 − cos ⁡ x x 2 = 1 2 {\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x^{2}}}={\frac {1}{2}}} lim x → n ± tan ⁡ ( π x + π 2 ) = ∓ ∞ {\displaystyle \lim _{x\to n^{\pm }}\tan \left(\pi x+{\frac {\pi }{2}}\right)=\mp \infty \qquad } যেকোনো অখণ্ড সংখ্যা n র জন্য। lim x → ∞ N / x = 0 {\displaystyle \lim _{x\to \infty }N/x=0} যেকোনো বাস্তব সংখ্যা N র জন্য। lim x → ∞ x / N = { ∞ , N > 0 does not exist , N = 0 − ∞ , N < 0 {\displaystyle \lim _{x\to \infty }x/N={\begin{cases}\infty ,&N>0\\{\text{does not exist}},&N=0\\-\infty ,&N<0\end{cases}}} lim x → ∞ x N = { ∞ , N > 0 1 , N = 0 0 , N < 0 {\displaystyle \lim _{x\to \infty }x^{N}={\begin{cases}\infty ,&N>0\\1,&N=0\\0,&N<0\end{cases}}} lim x → ∞ N x = { ∞ , N > 1 1 , N = 1 0 , 0 < N < 1 {\displaystyle \lim _{x\to \infty }N^{x}={\begin{cases}\infty ,&N>1\\1,&N=1\\0,&0<N<1\end{cases}}} lim x → ∞ N − x = lim x → ∞ 1 / N x = 0 {\displaystyle \lim _{x\to \infty }N^{-x}=\lim _{x\to \infty }1/N^{x}=0} যিকোনো N > 1 {\displaystyle N>1} র কারণে। lim x → ∞ N x = { 1 , N > 0 0 , N = 0 does not exist , N < 0 {\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{N}}={\begin{cases}1,&N>0\\0,&N=0\\{\text{does not exist}},&N<0\end{cases}}} lim x → ∞ x N = ∞ {\displaystyle \lim _{x\to \infty }{\sqrt[{N}]{x}}=\infty } যেকোনো N > 0 {\displaystyle N>0} র কারণে। lim x → ∞ log ⁡ x = ∞ {\displaystyle \lim _{x\to \infty }\log x=\infty } lim x → 0 + log ⁡ x = − ∞ {\displaystyle \lim _{x\to 0^{+}}\log x=-\infty }