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Category: Limits

lim-x-0-1-x-tan-pi-2-x-

Question Number 131181 by john_santu last updated on 02/Feb/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{{x}}\:−\:\mathrm{tan}\:\left(\frac{\pi}{\mathrm{2}}−{x}\right)\right)=? \\ $$ Answered by Ar Brandon last updated on 02/Feb/21 $$\mathscr{L}=\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{{x}}\:−\:\mathrm{tan}\:\left(\frac{\pi}{\mathrm{2}}−{x}\right)\right)=\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{\mathrm{x}}−\mathrm{cotx}\right) \\…

lim-x-x-2-cot-1-x-4cot-1-x-3x-2-2x-

Question Number 131156 by EDWIN88 last updated on 02/Feb/21 $$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}^{\mathrm{2}} \mathrm{cot}\:\left(\frac{\mathrm{1}}{{x}}\right)+\mathrm{4cot}\:\left(\frac{\mathrm{1}}{{x}}\right)}{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}{x}}\:? \\ $$ Answered by john_santu last updated on 02/Feb/21 $$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{{x}^{\mathrm{2}} +\mathrm{4}}{{x}\left(\mathrm{3}{x}+\mathrm{2}\right).\mathrm{tan}\:\left(\frac{\mathrm{1}}{{x}}\right)}\:=\:…

lim-h-0-e-2-2-x-e-2-2-h-e-2-2-x-h-

Question Number 131079 by mathlove last updated on 01/Feb/21 $$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{e}^{\mathrm{2}\sqrt{\mathrm{2}}{x}} {e}^{\mathrm{2}\sqrt{\mathrm{2}}{h}} −{e}^{\mathrm{2}\sqrt{\mathrm{2}}{x}} }{{h}}=? \\ $$ Commented by EDWIN88 last updated on 01/Feb/21 $$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{e}^{\mathrm{2}\sqrt{\mathrm{2}}\:\left({x}+{h}\right)}…

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} \\…