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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits3 Marks
Question
Evaluate the following limits:
$\lim _{x \rightarrow 0}\left[\frac{\sqrt[3]{1+x}-\sqrt{1+x}}{x}\right]$
✓

Answer

$ \lim _{x \rightarrow 0} \frac{\sqrt[3]{1+x}-\sqrt{1+x}}{x}$
$=\lim _{x \rightarrow 0} \frac{(1+x)^{\frac{1}{3}}-(1+x)^{\frac{1}{2}}}{x} $
Put $1+x=y$
As $x \rightarrow 0, y \rightarrow 1$
$ \lim _{x \rightarrow 0} \frac{(1+x)^{\frac{1}{3}}-(1+x)^{\frac{1}{2}}}{x}$
$=\lim _{y \rightarrow 1} \frac{y^{\frac{1}{3}}-y^{\frac{1}{2}}}{y-1}$
$=\lim _{y \rightarrow 1} \frac{\left(y^{\frac{1}{3}}-1\right)-\left(y^{\frac{1}{2}}-1\right)}{y-1}$
$=\lim _{y \rightarrow 1}\left(\frac{y^{\frac{1}{3}}-1}{y-1}-\frac{y^{\frac{1}{2}}-1}{y-1}\right)$
$=\lim _{y \rightarrow 1} \frac{y^{\frac{1}{3}}-1^{\frac{1}{3}}}{y-1}-\lim _{y \rightarrow 1} \frac{y^{\frac{1}{2}}-1^{\frac{1}{2}}}{y-1}$
$=\frac{1}{3}(1)^{\frac{-2}{3}}-\frac{1}{2}(1)^{\frac{-1}{2}} \quad \cdots\left[\because \lim _{x \rightarrow a} \frac{x^n-\mathrm{a}^n}{x-\mathrm{a}}=\mathrm{n} \cdot \mathrm{a}^{\mathrm{n}-1}\right]$
$=\frac{1}{3}-\frac{1}{2}$
$=\frac{2-3}{6}$
$=-\frac{1}{6}$

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