Question

Evaluate the following :

${\displaystyle \underset{x\to \text{a}}{lim}\left[\frac{\sin x-\sin \text{a}}{\sqrt[5]{x}-\sqrt[5]{\text{a}}}\right]}$

Answer 1

${\displaystyle \underset{x\to \text{a}}{lim}\frac{\sin x-\sin \text{a}}{\sqrt[5]{x}-\sqrt[5]{\text{a}}}}$

= ${\displaystyle \underset{x\to \text{a}}{lim}\frac{2\cos \left(\frac{x+\text{a}}{2}\right)\sin \left(\frac{x-\text{a}}{2}\right)}{\sqrt[5]{x}-\sqrt[5]{\text{a}}}}$

= ${\displaystyle \underset{x\to \text{a}}{lim}\frac{2\cos \left(\frac{x+\text{a}}{2}\right)\frac{\sin \left(\frac{x-\text{a}}{2}\right)}{\left(\frac{x-\text{a}}{2}\right)}}{\left(\frac{\sqrt[5]{x}-\sqrt[5]{\text{a}}}{x-\text{a}}\right)}\times \frac{1}{2} ...\left[\begin{array}{c}\because x\to \text{a}\text{,} x\ne \text{a}\\ \therefore x-\text{a}\ne 0\end{array}\right]}$

= ${\displaystyle \frac{\left[\underset{x\to \text{a}}{lim}\cos \left(\frac{x+\text{a}}{2}\right)\right]\times \left[\underset{x\to \text{a}}{lim}\frac{\sin \left(\frac{x-\text{a}}{2}\right)}{\left(\frac{x-\text{a}}{2}\right)}\right]}{\underset{x\to \text{a}}{lim}\left(\frac{x^{\frac{1}{5}}-{\text{a}}^{\frac{1}{5}}}{x-\text{a}}\right)}}$

= ${\displaystyle \frac{\cos \left(\frac{\text{a}+\text{a}}{2}\right)\cdot 1}{\frac{1}{5}{\text{a}}^{-\frac{4}{5}}} ...\left[\begin{array}{c}\because x\to \text{a}\text{,} x\ne \text{a} \therefore \frac{x-\text{a}}{2}\to 0\\ \text{and}\underset{\theta \to 0}{lim}\frac{\sin \theta }{\theta }=1\\ \text{Also}\text{,}\underset{x\to \text{a}}{lim}\frac{x^{\text{n}}-{\text{a}}^{\text{n}}}{x-\text{a}}={\text{na}}^{\text{n}-1}\end{array}\right]}$

= ${\displaystyle 5{\text{a}}^{\frac{4}{5}}\cos \text{a}}$