# 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
$${let}\:{f}\left({x}\right)=\frac{\mathrm{1}−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{cosx}}{{x}^{\mathrm{4}} } \\$$$${we}\:{hsve}\:{for}\:{x}\in{V}\left(\mathrm{0}\right)\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\sim\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}} \\$$$${and}\:{cosx}\:\sim\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\:\Rightarrow\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{cosx}\:\sim\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right)\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\right) \\$$$$=\mathrm{1}−\frac{{x}^{\mathrm{4}} }{\mathrm{4}}\:\Rightarrow\mathrm{1}−\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{cosx}\:\sim\frac{{x}^{\mathrm{4}} }{\mathrm{4}}\:\Rightarrow\:{f}\left({x}\right)\sim\frac{\mathrm{1}}{\mathrm{4}}\:\Rightarrow \\$$$${lim}_{{x}\rightarrow\mathrm{0}} {f}\left({x}\right)=\frac{\mathrm{1}}{\mathrm{4}} \\$$
Commented by john santu last updated on 27/Dec/19
$${yes}\:{sir}\:{i}\:{got}\:{the}\:{same}\:{result} \\$$
Commented by benjo last updated on 27/Dec/19
$$\mathrm{waw}…\mathrm{i}\:\mathrm{like}\:\mathrm{this}\:\mathrm{question}.\:\mathrm{limit} \\$$$$\mathrm{afdoll} \\$$
Commented by john santu last updated on 13/Jan/20
$${sir}\:{this}\:{answer}\:{not}\:{correct}.\: \\$$$${value}\:{of}\:{limit}\:=\:\frac{\mathrm{1}}{\mathrm{3}} \\$$