how-evaluate-lim-x-0-1-1-x-2-cos-x-x-4-

Question Number 76445 by john santu last updated on 27/Dec/19

$${how}\:{evaluate}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }×\mathrm{cos}\:{x}}{{x}^{\mathrm{4}} }\right). \\ $$

Commented by JDamian last updated on 27/Dec/19

$${What}\:{if}\:{you}\:{use}\:{L}'{Hopital}'{s}\:{rule}? \\ $$

Commented by john santu last updated on 27/Dec/19

$${reques}\:{Taylor}\:{series}\:{sir} \\ $$

Commented by abdomathmax last updated on 27/Dec/19

let f(x)=((1−(√(1+x^2 ))cosx)/x^4 ) we hsve for x∈V(0) (√(1+x^2 ))∼1+(x^2 /2) and cosx ∼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) ⇒ f(x)∼(1/4) ⇒ lim_(x→0) f(x)=(1/4)

$${let}\:{f}\left({x}\right)=\frac{\mathrm{1}−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{cosx}}{{x}^{\mathrm{4}} } \\ $$$${we}\:{hsve}\:{for}\:{x}\in{V}\left(\mathrm{0}\right)\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\sim\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}} \\ $$$${and}\:{cosx}\:\sim\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\:\Rightarrow\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{cosx}\:\sim\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right) \\ $$$$=\mathrm{1}−\frac{{x}^{\mathrm{4}} }{\mathrm{4}}\:\Rightarrow\mathrm{1}−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{cosx}\:\sim\frac{{x}^{\mathrm{4}} }{\mathrm{4}}\:\Rightarrow\:{f}\left({x}\right)\sim\frac{\mathrm{1}}{\mathrm{4}}\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{0}} {f}\left({x}\right)=\frac{\mathrm{1}}{\mathrm{4}} \\ $$

Commented by john santu last updated on 27/Dec/19