Category: Limits

Question Number 122721 by bemath last updated on 19/Nov/20 $$\underset{x\to 1}{\lim }{\left(\frac{1+x}{2+x}\right)}^{\frac{1-\sqrt{x}}{1-x}}=?$$ Answered by Dwaipayan Shikari last updated on 19/Nov/20 $$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\left(\frac{\mathrm{1}+{x}}{\mathrm{2}+{x}}\right)^{\frac{\mathrm{1}−\sqrt{{x}}}{\left(\mathrm{1}+\sqrt{{x}}\right)\left(\mathrm{1}−\sqrt{{x}}\right)}} \…

Question Number 122624 by Ar Brandon last updated on 18/Nov/20 $$\mathrm{Prove}\mathrm{the}\mathrm{equality}:$$$$\mathrm{n}!=\underset{\mathrm{k}=0}{\sum ^{\mathrm{n}}}{(-1)}^{\mathrm{k}}\left(\begin{array}{c}\mathrm{n}\\ \mathrm{k}\end{array}\right){(\mathrm{n}-\mathrm{k})}^{\mathrm{n}}$$ Answered by mindispower last updated on 18/Nov/20…

Question Number 57073 by gunawan last updated on 31/Mar/19 $$f\left(x\right)=A$$$$f:[0,\mathrm{\infty })$$$$g:[1,0]$$$$g\left(x\right)=B$$$$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}{\int }_0^{n}f\left(x\right)g\left(\frac{x}{n}\right)dx=\dots$$ Commented by…

Question Number 122548 by bramlexs22 last updated on 18/Nov/20 $$\underset{x\to 1/2}{\lim }\left(\frac{\cot \left(\pi x\right)}{2x^2+x-1}\right)=?$$ Answered by liberty last updated on 18/Nov/20 $$\:\underset{{x}\rightarrow\mathrm{1}/\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{cot}\:\left(\pi\mathrm{x}\right)}{\left(\mathrm{2x}−\mathrm{1}\right)\left(\mathrm{x}+\mathrm{1}\right)}\:=\:\underset{{x}\rightarrow\mathrm{1}/\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{x}+\mathrm{1}}.\underset{\left(\frac{\mathrm{2x}−\mathrm{1}}{\mathrm{2}}\right)\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{cot}\:\left(\pi\mathrm{x}\right)}{\left(\mathrm{2x}−\mathrm{1}\right)}…

Question Number 122462 by benjo_mathlover last updated on 17/Nov/20 $$\underset{x\to \mathrm{\infty }}{\lim }\frac{\tan x}{x}?$$ Answered by Olaf last updated on 17/Nov/20 $$undeterminate$$ Commented by…