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Evaluate   \mathrm{\lim _{x \rightarrow 0} \frac{\ln (1+\sqrt{x \times \sin (x)})}{x}}

Option: 1

0


Option: 2

-1


Option: 3

1


Option: 4

2


Answers (1)

best_answer

\mathrm{\lim _{x \rightarrow 0^{+}} \frac{\ln (1+\sqrt{x \sin x})}{x}=\lim _{x \rightarrow 0^{+}} \frac{\ln (1+\sqrt{x \sin x})}{\sqrt{x^{2}}} }
                                               \mathrm{ =\lim _{x \rightarrow 0^{+}} \sqrt{\frac{\sin x}{x} \times \frac{1}{x \sin x}} \ln (1+\sqrt{x \sin x}) }
                                               \mathrm{ =\lim _{x \rightarrow 0^{+}} \sqrt{\frac{\sin x}{x}} \frac{\ln (1+\sqrt{x \sin x})}{\sqrt{x \sin x}} }
                                               \mathrm{ = \sqrt{1}\times 1}

 
\mathrm{ \lim _{x \rightarrow 0^{+}} \frac{\ln (1+\sqrt{x \sin x})}{x}=1 }
  \mathrm{ \lim _{x \rightarrow 0^{-}} \frac{\ln (1+\sqrt{x \sin x})}{x}=\lim _{x \rightarrow 0^{-}} \frac{\ln (1+\sqrt{x \sin x})}{-\sqrt{x^{2}}} } 
                                               \mathrm{ =\lim _{x \rightarrow 0^{-}}-\sqrt{\frac{\sin x}{x} \times \frac{1}{x \sin x}} \ln (1+\sqrt{x \sin x}) }
                                              \mathrm{ =\lim _{x \rightarrow 0^{-}}-\sqrt{\frac{\sin x}{x}} \frac{\ln (1+\sqrt{x \sin x})}{\sqrt{x \sin x}} }
                                              \mathrm{ =-\sqrt{1} \times 1 }

\mathrm{ \lim _{x \rightarrow 0^{-}} \frac{\ln (1+\sqrt{x \sin x})}{x}=-1 }
Because \mathrm{ \lim _{x \rightarrow 0} \frac{\ln (1+\sqrt{x \sin x})}{\sqrt{x \sin x}}=\lim _{y \rightarrow 0} \frac{\ln (1+y)}{y}=1}.

Posted by

Ritika Kankaria

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