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

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

cosx∼1−(x^2 /2) and (√(1+x^2 ))∼1+(x^2 /2) ⇒(√(1+x^2 ))cosx∼(1−(x^2 /2))(1+(x^2 /2)) =1−(x^4 /4) ⇒1−(√(1+x^2 ))cosx∼(x^4 /4) and tan^4 x∼ x^4 ⇒ lim_(x→0) ((1−(√(1+x^2 ))cosx)/(tan^4 x))=(1/4)

$$\mathrm{cosx}\sim\mathrm{1}−\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}\:\mathrm{and}\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\sim\mathrm{1}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}\:\Rightarrow\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{cosx}\sim\left(\mathrm{1}−\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}\right)\left(\mathrm{1}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}\right) \\ $$$$=\mathrm{1}−\frac{\mathrm{x}^{\mathrm{4}} }{\mathrm{4}}\:\Rightarrow\mathrm{1}−\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{cosx}\sim\frac{\mathrm{x}^{\mathrm{4}} }{\mathrm{4}}\:\:\mathrm{and}\:\mathrm{tan}^{\mathrm{4}} \mathrm{x}\sim\:\mathrm{x}^{\mathrm{4}} \:\Rightarrow \\ $$$$\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \frac{\mathrm{1}−\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{cosx}}{\mathrm{tan}^{\mathrm{4}} \mathrm{x}}=\frac{\mathrm{1}}{\mathrm{4}} \\ $$

Commented by bobhans last updated on 15/Jun/21