Question:
Evaluate
$\lim _{x \rightarrow a}\left(\frac{\sqrt{x}-\sqrt{a}}{x-a}\right)$
Solution:
To evaluate:
$\lim _{x \rightarrow a} \frac{\sqrt{x}-\sqrt{a}}{x-a}$
Formula used:
We have,
$\frac{x^{m}-y^{m}}{x-y}=m y^{m-1}$
As $\mathrm{x} \rightarrow \mathrm{a}$, we have
$\lim _{x \rightarrow a} \frac{x^{\frac{1}{2}}-a^{\frac{1}{2}}}{x-a}=\frac{1}{2} a^{\frac{1}{2}-1}$
$\lim _{x \rightarrow a} \frac{x^{\frac{1}{2}}-a^{\frac{1}{2}}}{x-a}=\frac{1}{2 \sqrt{a}}$
Thus, the value of $\lim _{x \rightarrow a} \frac{x^{\frac{1}{2}}-a^{\frac{1}{2}}}{x-a}$ is $\frac{1}{2 \sqrt{a}}$