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Question Number 115167 by bemath last updated on 28/Sep/20

   lim_(x→0)  ((sec  (sin^(−1) (1−x)))/(3(√x))) = ?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sec}\:\:\left(\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{1}−{x}\right)\right)}{\mathrm{3}\sqrt{{x}}}\:=\:? \\ $$

Answered by bobhans last updated on 28/Sep/20

 lim_(x→0)  ((sec  (sin^(−1) (1−x)))/(3(√x))) =?  remaining sec  (sin^(−1) (1−x)) = (√(2x−x^2 ))  so the limit can write as   lim_(x→0)  ((√(2x−x^2 ))/(3(√x))) = lim_(x→0)  (((√x) .(√(2−x)))/(3(√x))) = ((√2)/3)

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sec}\:\:\left(\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{1}−{x}\right)\right)}{\mathrm{3}\sqrt{{x}}}\:=? \\ $$$${remaining}\:\mathrm{sec}\:\:\left(\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{1}−{x}\right)\right)\:=\:\sqrt{\mathrm{2}{x}−{x}^{\mathrm{2}} } \\ $$$${so}\:{the}\:{limit}\:{can}\:{write}\:{as}\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2}{x}−{x}^{\mathrm{2}} }}{\mathrm{3}\sqrt{{x}}}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{{x}}\:.\sqrt{\mathrm{2}−{x}}}{\mathrm{3}\sqrt{{x}}}\:=\:\frac{\sqrt{\mathrm{2}}}{\mathrm{3}} \\ $$

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