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Question Number 121074 by bramlexs22 last updated on 05/Nov/20

 lim_(x→1)  ((x−1)/( (√(x^2 −1))))?

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

Answered by liberty last updated on 05/Nov/20

 lim_(x→1)  ((x−1)/( (√((x−1)(x+1))))) = lim_(x→1)  (√((x−1)/(x+1))) = 0

$$\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{x}−\mathrm{1}}{\:\sqrt{\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}+\mathrm{1}\right)}}\:=\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\sqrt{\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}+\mathrm{1}}}\:=\:\mathrm{0} \\ $$

Answered by liberty last updated on 05/Nov/20

L′Hopital    lim_(x→1)  ((x−1)/( (√(x^2 −1)))) = lim_(x→1)  (1/([(x/( (√(x^2 −1))))]))  = lim_(x→1)  ((√(x^2 −1))/x) = 0

$$\mathrm{L}'\mathrm{Hopital}\: \\ $$$$\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{x}−\mathrm{1}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}}\:=\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{1}}{\left[\frac{\mathrm{x}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}}\right]} \\ $$$$=\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}}{\mathrm{x}}\:=\:\mathrm{0} \\ $$

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