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Category: Limits

lim-x-pi-4-pi-4x-1-sin-2x-

Question Number 143995 by liberty last updated on 20/Jun/21 $$\:\:\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\pi−\mathrm{4}{x}}{\:\sqrt{\mathrm{1}−\sqrt{\mathrm{sin}\:\mathrm{2}{x}}}}\:=? \\ $$ Answered by mathmax by abdo last updated on 20/Jun/21 $$\mathrm{f}\left(\mathrm{x}\right)=\frac{\pi−\mathrm{4x}}{\:\sqrt{\mathrm{1}−\sqrt{\mathrm{sin2x}}}}\:\Rightarrow\mathrm{f}\left(\mathrm{x}\right)=_{\frac{\pi}{\mathrm{4}}−\mathrm{x}=\mathrm{t}} \:\:\:\frac{\pi−\mathrm{4}\left(\frac{\pi}{\mathrm{4}}−\mathrm{t}\right)}{\:\sqrt{\mathrm{1}−\sqrt{\mathrm{sin}\left(\mathrm{2}\left(\frac{\pi}{\mathrm{4}}−\mathrm{t}\right)\right.}}} \\…

The-value-of-lim-x-0-1-cos-x-2-1-cos-x-

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

lim-x-0-1-cos-x-x-is-equals-to-

Question Number 12881 by kunalshukla95040 last updated on 05/May/17 $$\frac{{lim}}{{x}\rightarrow\mathrm{0}}\frac{\sqrt{\mathrm{1}−\mathrm{cos}\:{x}}}{{x}} \\ $$$${is}\:{equals}\:{to}. \\ $$ Answered by nume1114 last updated on 05/May/17 $$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt{\mathrm{1}−\mathrm{cos}\:{x}}}{{x}} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}}…

Question-78392

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

lim-x-0-1-sin-4x-cos-2x-x-1-1-x-

Question Number 78324 by jagoll last updated on 16/Jan/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}+\mathrm{sin}\:\mathrm{4}{x}−\mathrm{cos}\:\mathrm{2}{x}}{\:\sqrt{{x}+\mathrm{1}}−\sqrt{\mathrm{1}−{x}}}= \\ $$ 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}}…