Question Number 143532 by bramlexs22 last updated on 15/Jun/21
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:\mathrm{cos}\:{x}}{\mathrm{tan}\:^{\mathrm{4}} {x}}\:=? \\ $$
Answered by mathmax by abdo last updated on 15/Jun/21
$$\mathrm{cosx}\sim\mathrm{1}−\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}\:\mathrm{and}\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\sim\mathrm{1}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}\:\Rightarrow\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{cosx}\sim\left(\mathrm{1}−\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}\right)\left(\mathrm{1}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}\right) \\ $$$$=\mathrm{1}−\frac{\mathrm{x}^{\mathrm{4}} }{\mathrm{4}}\:\Rightarrow\mathrm{1}−\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{cosx}\sim\frac{\mathrm{x}^{\mathrm{4}} }{\mathrm{4}}\:\:\mathrm{and}\:\mathrm{tan}^{\mathrm{4}} \mathrm{x}\sim\:\mathrm{x}^{\mathrm{4}} \:\Rightarrow \\ $$$$\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \frac{\mathrm{1}−\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{cosx}}{\mathrm{tan}^{\mathrm{4}} \mathrm{x}}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$
Commented by bobhans last updated on 15/Jun/21
$${i}\:{think}\:{the}\:{answer}\:{is}\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$
Commented by Mathspace last updated on 15/Jun/21
$${show}\:{your}\:{work}\:{sir} \\ $$
Commented by bobhans last updated on 15/Jun/21
