Find the value

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 $\frac{1}{2 \sqrt{a}}$

 

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