Question:
Evaluate
$\lim _{x \rightarrow a}\left(\frac{x^{5 / 2}-a^{5 / 2}}{x-a}\right)$
Solution:
To evaluate:
$\lim _{x \rightarrow a} \frac{x^{\frac{5}{2}}-a^{\frac{5}{2}}}{x-a}$
Formula used: We have,
$\frac{x^{m}-y^{m}}{x-y}=m y^{m-1}$
As $x \rightarrow$ a, we have
$\lim _{x \rightarrow a} \frac{x^{\frac{5}{2}}-a^{\frac{5}{2}}}{x-a}=\frac{5}{2} a^{\frac{5}{2}-1}$
$\lim _{x \rightarrow a} \frac{x^{\frac{5}{2}}-a^{\frac{5}{2}}}{x-a}=\frac{5}{2} a^{\frac{3}{2}}$
Thus, the value of $\lim _{x \rightarrow a} \frac{x^{\frac{5}{2}}-a^{\frac{5}{2}}}{x-a}$ is $\frac{5}{2} a^{\frac{3}{2}}$