Category: Limits

Question Number 143960 by bobhans last updated on 20/Jun/21 $$\mathrm{The}\mathrm{value}\mathrm{of}\underset{x\to 0}{\lim }\frac{\sqrt{1-\cos {\mathrm{x}}^2}}{1-\cos \mathrm{x}}=?$$ Answered by lapache last updated on 20/Jun/21 $${li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\frac{\sqrt{\mathrm{1}−\mathrm{1}+\frac{{x}^{\mathrm{4}} }{\mathrm{2}}\:}}{\mathrm{1}−\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}}={li}\underset{{x}\rightarrow\mathrm{0}}…

Question Number 78392 by jagoll last updated on 17/Jan/20 Answered by john santu last updated on 17/Jan/20 $$leth=\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}\sqrt[4]{\cos 4x}\dots \sqrt[n]{\cos nx}$$$$ln\left(h\right)=ln\left(\cos x\right)+\frac{1}{2}ln\left(\cos 2x\right)+\frac{1}{3}ln\left(\cos 3x\right)+\dots +\frac{1}{n}ln\left(\cos nx\right)$$$$\frac{dh}{dx}=\frac{-\sin x}{\cos x}-\frac{\sin 2x}{\cos 2x}-\frac{\sin 3x}{\cos 3x}-\dots -\frac{\sin nx}{\cos nx}$$$${now}\:{we}\:{use}\:{L}'{Hopital}\:{rule} \…

Question Number 78324 by jagoll last updated on 16/Jan/20 $$\underset{x\to 0}{\lim }\frac{1+\sin 4x-\cos 2x}{\sqrt{x+1}-\sqrt{1-x}}=$$ Commented by john santu last updated on 16/Jan/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\sqrt{{x}+\mathrm{1}}+\sqrt{\mathrm{1}−{x}}\right)×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2sin}\:^{\mathrm{2}} {x}+\mathrm{2sin}\:\mathrm{2}{x}\mathrm{cos}\:\mathrm{2}{x}}{\mathrm{2}{x}}…

Question Number 12732 by Joel577 last updated on 30/Apr/17 $$\underset{x\to 0}{\lim }\frac{\sqrt{x}-x}{\sqrt{x}+x}$$ Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 30/Apr/17 $$=1$$ Commented by…

Question Number 143677 by bobhans last updated on 17/Jun/21 Answered by bemath last updated on 17/Jun/21 Answered by mathmax by abdo last updated on 17/Jun/21…