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Question Number 106313 by bemath last updated on 04/Aug/20

lim_(x→1)  ((x∣x−1∣)/(x^2 −1))

$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{x}\mid\mathrm{x}−\mathrm{1}\mid}{\mathrm{x}^{\mathrm{2}} −\mathrm{1}} \\ $$

Answered by Dwaipayan Shikari last updated on 04/Aug/20

first case  lim_(x→1^+ ) ((x(x−1))/(x^2 −1))=(x/(x+1))=(1/2)  second case  lim_(x→1) −((x(x−1))/(x^2 −1))=−(1/2)  L.H.S≠R.H.S(limit doesn′t exist)

$${first}\:{case} \\ $$$$\underset{{x}\rightarrow\mathrm{1}^{+} } {\mathrm{lim}}\frac{{x}\left({x}−\mathrm{1}\right)}{{x}^{\mathrm{2}} −\mathrm{1}}=\frac{{x}}{{x}+\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${second}\:{case} \\ $$$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}−\frac{{x}\left({x}−\mathrm{1}\right)}{{x}^{\mathrm{2}} −\mathrm{1}}=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${L}.{H}.{S}\neq{R}.{H}.{S}\left({limit}\:{doesn}'{t}\:{exist}\right) \\ $$

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