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Limits — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsLimits4 Marks
Question
Evaluate the following limits: $\lim _{x \rightarrow a} \frac{\sin x-\sin a}{\sqrt[5]{x}-\sqrt[5]{a}}$
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Answer

$\lim _{x \rightarrow \mathrm{a}} \frac{\sin x-\sin \mathrm{a}}{\sqrt[5]{x}-\sqrt[5]{\mathrm{a}}}$
$=\lim _{x \rightarrow \mathrm{a}} \frac{2 \cos \left(\frac{x+\mathrm{a}}{2}\right) \cdot \sin \left(\frac{x-\mathrm{a}}{2}\right)}{x^{\frac{1}{5}}-\mathrm{a}^{\frac{1}{5}}}$
$=\lim _{x \rightarrow a} \frac{2 \cos \left(\frac{x+a}{2}\right) \cdot \frac{\sin \left(\frac{x-a}{2}\right)}{x-a}}{\frac{x^{\frac{1}{5}}-a^{\frac{1}{5}}}{x-a}}\cdots\left[\begin{array}{l}
\text { Divide numerator and denominator by } x-\mathrm{a} . \\ \because x \rightarrow \mathrm{a}, x \neq \mathrm{a}, \therefore x-\mathrm{a} \neq 0 \end{array}\right]$
$=\frac{\lim _{x \rightarrow \mathrm{a}} \cos \left(\frac{x+\mathrm{a}}{2}\right) \cdot \lim _{x \rightarrow \mathrm{a}}\left[\frac{\sin \left(\frac{x-\mathrm{a}}{2}\right)}{\frac{x-\mathrm{a}}{2}}\right]}{\lim _{x \rightarrow \mathrm{a}} \frac{x^{\frac{1}{5}}-\mathrm{a}^{\frac{1}{5}}}{x-\mathrm{a}}}$
$=\frac{\cos \left(\frac{a+a}{2}\right) \times 1}{\frac{1}{5} \cdot a^{\frac{-4}{5}}}$
$\ldots\left[\begin{array}{l} \because x \rightarrow a, x-a \rightarrow 0 \\ \therefore \frac{x-a}{2} \rightarrow 0 ; \lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1 \\ \text { and } \lim _{x \rightarrow 0} \frac{x^n-a^n}{x-a}=n a^{n-1}
\end{array}\right]$
$=5 a^{\frac{4}{5}} \cdot \cos a$

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