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  • Is the function <img alt="\mathrm{\frac{\sqrt{(1+x)}-\sqrt{(1-x)}}{x}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cfrac%7B%5Csqrt%7B%281&plus;x%29%7D-%5Csqrt%7B%281-x%2

Is the function \mathrm{\frac{\sqrt{(1+x)}-\sqrt{(1-x)}}{x}}defined for all values of x ? Indicate the values of x for which it is defined and real. Find the limit as \mathrm{ x \rightarrow 0.}

Option: 1

2


Option: 2

1


Option: 3

3


Option: 4

4


Answers (1)

best_answer

The given function is not defined at \mathrm{x=0} since it takes the indeterminate form \mathrm{0 / 0} at \mathrm{x=0}. Also

\mathrm{1+x \geq 0,1-x \geq 0} or  \mathrm{-1 \leq x \leq 1}. The value of the function is imaginary for all values of \mathrm{x>1} and \mathrm{x<-1}

and so it is also not defined for these values of x. It follows that the function is defined and real for all values of x in the interval

\mathrm{-1 \leq x \leq 1 \, \, except x=0}. Now

\mathrm{ \begin{aligned} & \lim _{x \rightarrow 0} \frac{\sqrt{(1+x)}-\sqrt{(1-x)}}{x} \\ & =\lim _{x \rightarrow 0} \frac{(1+x)-(1-x)}{x[\sqrt{ }(1+x)+\sqrt{ }(1-x)]} \\ & =\lim _{x \rightarrow 0} \frac{2}{\sqrt{(1+x)+\sqrt{ }(1-x)}}=\frac{2}{1+1}=1 . \\ & \end{aligned} }

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Divya Prakash Singh

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