Category: Limits

Question Number 14336 by tawa tawa last updated on 30/May/17 Commented by ajfour last updated on 30/May/17 $$\cos x=1-(1-\cos x)$$$$=1-2\sin {}^2\left(\frac{x}{2}\right)=1-t$$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{cos}\:{x}\right)=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left[\left(\mathrm{1}−{t}\right)^{−\mathrm{1}/{t}}…

Question Number 145205 by imjagoll last updated on 03/Jul/21 $$\underset{x\to 0}{\lim }\frac{1-\cos 2\left(\sin \left(\sin \mathrm{x}\right)\right)}{{\mathrm{x}}^2}=?$$ Answered by EDWIN88 last updated on 03/Jul/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{2}\left(\mathrm{sin}\:\left(\mathrm{sin}\:{x}\right)\right)}{{x}^{\mathrm{2}} }\: \…

Question Number 145129 by liberty last updated on 02/Jul/21 Answered by EDWIN88 last updated on 03/Jul/21 $$\:\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:−\mathrm{1}\right)}{\mathrm{cos}\:^{\mathrm{2}} \left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:−\mathrm{1}\right)\left(\mathrm{2sin}^{\mathrm{2}} \left(\frac{\sqrt{\mathrm{2}{x}}}{\mathrm{2}}\right)\right)^{{n}} \:}\:=\:{a}…

Question Number 145065 by Ar Brandon last updated on 02/Jul/21 $$\frac{1}{\sqrt{1+\mathrm{x}}}=1+\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}\left[\frac{{(-1)}^{\mathrm{k}}}{2^{2\mathrm{k}}}{\mathrm{C}}_{2\mathrm{k}}^{\mathrm{k}}\right]{\mathrm{x}}^{\mathrm{k}}+\mathrm{o}\left({\mathrm{x}}^{\mathrm{n}}\right)$$ Terms of Service Privacy…

Question Number 145012 by mim24 last updated on 01/Jul/21 Answered by ArielVyny last updated on 01/Jul/21 $$lim\left(x^{-\frac{1}{x}}\right)(x\to 0)$$$$lim\left(e^{-\frac{1}{x}lnx}\right)(x\to 0^{+})=+\mathrm{\infty }$$$$ \…