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Category: Limits

1-lim-x-0-1-cos-6-2x-cos-3-3x-3x-2-2-lim-x-0-1-cos-4x-2sin-2-x-cos-4x-x-2-cos-3x-3-lim-x-pi-2-sin-x-2cos-2-x-1-sin-3-x-sin-x-

Question Number 115320 by john santu last updated on 25/Sep/20 $$\left(\mathrm{1}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:^{\mathrm{6}} \left(\mathrm{2}{x}\right)\mathrm{cos}\:^{\mathrm{3}} \left(\mathrm{3}{x}\right)}{\mathrm{3}{x}^{\mathrm{2}} }\:? \\ $$$$\left(\mathrm{2}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{4}{x}+\mathrm{2sin}\:^{\mathrm{2}} {x}.\mathrm{cos}\:\mathrm{4}{x}}{{x}^{\mathrm{2}} .\mathrm{cos}\:\mathrm{3}{x}}? \\ $$$$\left(\mathrm{3}\right)\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{2}}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:{x}−\mathrm{2cos}\:^{\mathrm{2}} {x}−\mathrm{1}}{\:\sqrt{\mathrm{sin}\:^{\mathrm{3}} {x}}−\sqrt{\mathrm{sin}\:{x}}}\:?\:…

lim-x-0-xsin-x-2sin-2-3x-x-2-cos-x-

Question Number 115318 by bemath last updated on 25/Sep/20 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}\mathrm{sin}\:{x}}{\mathrm{2sin}\:^{\mathrm{2}} \left(\mathrm{3}{x}\right)−{x}^{\mathrm{2}} \mathrm{cos}\:{x}} \\ $$ Answered by bobhans last updated on 25/Sep/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}\:\mathrm{sin}\:{x}}{\mathrm{2}\:\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{3}{x}\right)−{x}^{\mathrm{2}}…

lim-x-pi-8-cot-4x-cos-4x-pi-8x-3-

Question Number 115195 by bemath last updated on 24/Sep/20 $$\underset{{x}\rightarrow\frac{\pi}{\mathrm{8}}} {\mathrm{lim}}\:\frac{\mathrm{cot}\:\mathrm{4}{x}−\mathrm{cos}\:\mathrm{4}{x}}{\left(\pi−\mathrm{8}{x}\right)^{\mathrm{3}} }\:?\: \\ $$ Answered by bobhans last updated on 24/Sep/20 $${let}\:{x}\:=\:\frac{\pi}{\mathrm{8}}+{p}\:;\:{p}\rightarrow\mathrm{0} \\ $$$$\underset{{p}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cot}\:\left(\mathrm{4}{p}+\frac{\pi}{\mathrm{2}}\right)−\mathrm{cos}\:\left(\mathrm{4}{p}+\frac{\pi}{\mathrm{2}}\right)}{\left(−\mathrm{8}{p}\right)^{\mathrm{3}}…

lim-x-1-tan-cos-1-1-x-x-1-

Question Number 115174 by bemath last updated on 24/Sep/20 $$\:\:\:\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\left(\mathrm{cos}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{{x}}\right)\right)}{\:\sqrt{{x}−\mathrm{1}}}\:=\:? \\ $$ Answered by bobhans last updated on 24/Sep/20 $$\:{let}\:\mathrm{cos}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{{x}}\right)\:=\:\psi\:\Rightarrow\:\frac{\mathrm{1}}{{x}}\:=\:\mathrm{cos}\:\psi\: \\ $$$$\:{and}\:\mathrm{tan}\:\psi\:=\:\sqrt{{x}^{\mathrm{2}}…

lim-x-0-sec-sin-1-1-x-3-x-

Question Number 115167 by bemath last updated on 28/Sep/20 $$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sec}\:\:\left(\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{1}−{x}\right)\right)}{\mathrm{3}\sqrt{{x}}}\:=\:? \\ $$ Answered by bobhans last updated on 28/Sep/20 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sec}\:\:\left(\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{1}−{x}\right)\right)}{\mathrm{3}\sqrt{{x}}}\:=? \\…

1-lim-x-1-arc-sin-1-x-1-x-2-lim-x-e-x-3-cos-1-x-2-3-lim-x-0-csc-x-sin-sin-x-

Question Number 115162 by john santu last updated on 24/Sep/20 $$\:\left(\mathrm{1}\right)\:\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\mathrm{arc}\:\mathrm{sin}\:\left(\frac{\mathrm{1}−\sqrt{{x}}}{\mathrm{1}−{x}}\right)\:=? \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{e}^{{x}^{\mathrm{3}} +\sqrt{\mathrm{cos}\:\left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)}} \:=? \\ $$$$\left(\mathrm{3}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\mathrm{csc}\:{x}\:.\mathrm{sin}\:\left(\mathrm{sin}\:{x}\right)\:=? \\ $$ Answered by…