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Continuity — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsContinuity3 Marks
Question
Find f(x) is continuse at x = 0, then $\text{f(x)}=\frac{\text{x}}{1-\sqrt{1-\text{x}}}$ becomes continuous at x = 0.
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Answer

If f(x) is continuous at x = 0, then $\lim\limits_{{\text{x}}\rightarrow0}\text{f(x})=\text{f}(0)\ ....(\text{i})$
Given, $\text{f(x)}=\frac{\text{x}}{1-\sqrt{1-\text{x}}}$
$\Rightarrow\text{f(x)}=\frac{\text{x}\big(1+\sqrt{1-\text{x}}\big)}{\big(1-\sqrt{1-\text{x}}\big)\big(1+\sqrt{1-\text{x}}\big)}$
$\Rightarrow\text{f(x)}=\frac{\text{x}\big(1+\sqrt{1-\text{x}}\big)}{1-(1-\text{x})}$
$\Rightarrow\text{f(x)}=\big(1+\sqrt{1-\text{x}}\big)$
$\Rightarrow\lim\limits_{{\text{x}}\rightarrow0}=\big(1+\sqrt{1-\text{x}}\big)=\text{f}(0)$ [From eq. (i)]
$\Rightarrow\text{f}(0)=2$
So, for f(0) = 2, the function f(x) becomes continuous x = 0

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