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Question Number 59694 by meme last updated on 13/May/19

lim_(x→0) ((cos((√(∣x∣)−1)))/x)=?

$${li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\frac{{cos}\left(\sqrt{\left.\mid{x}\mid\right)−\mathrm{1}}\right.}{{x}}=? \\ $$

Commented by maxmathsup by imad last updated on 13/May/19

let A(x) =((cos((√(∣x∣)))−1)/x)    we have cos(u) ∼1−(u^2 /2) (  u∈V(0)) ⇒  cos((√(∣x∣)))∼1−((∣x∣)/2) ⇒cos((√(∣x∣)))−1  ∼−((∣x∣)/2)    now its clear that  lim_(x→0^+ )    A(x) =−(1/2)   and  lim_(x→0^− )    A(x) =(1/2)

$${let}\:{A}\left({x}\right)\:=\frac{{cos}\left(\sqrt{\mid{x}\mid}\right)−\mathrm{1}}{{x}}\:\:\:\:{we}\:{have}\:{cos}\left({u}\right)\:\sim\mathrm{1}−\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:\left(\:\:{u}\in{V}\left(\mathrm{0}\right)\right)\:\Rightarrow \\ $$$${cos}\left(\sqrt{\mid{x}\mid}\right)\sim\mathrm{1}−\frac{\mid{x}\mid}{\mathrm{2}}\:\Rightarrow{cos}\left(\sqrt{\mid{x}\mid}\right)−\mathrm{1}\:\:\sim−\frac{\mid{x}\mid}{\mathrm{2}}\:\:\:\:{now}\:{its}\:{clear}\:{that} \\ $$$${lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:\:{A}\left({x}\right)\:=−\frac{\mathrm{1}}{\mathrm{2}}\:\:\:{and}\:\:{lim}_{{x}\rightarrow\mathrm{0}^{−} } \:\:\:{A}\left({x}\right)\:=\frac{\mathrm{1}}{\mathrm{2}} \\ $$

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