Evaluate: _ x a (2+x)^5 2 -(a+2)^5 2 x-a - Vidyadip

Question

Evaluate:
$\lim\limits_{\text{x} \rightarrow{\text{a}}}\frac{(2+\text{x})^\frac{5}{2}-(\text{a}+2)^\frac{5}{2}}{\text{x}-\text{a}}$

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Answer

Given that $\lim\limits_{\text{x} \rightarrow{\text{a}}}\frac{(2+\text{x})^\frac{5}{2}-(\text{a}+2)^\frac{5}{2}}{\text{x}-\text{a}}$
$=\lim\limits_{\text{x} \rightarrow{\text{a}}}\frac{(2+\text{x})^\frac{5}{2}-(\text{a}+2)^\frac{5}{2}}{(2+\text{x})-(\text{a}+2)}$
$=\lim\limits_{2+\text{x} \rightarrow \text{a}+2}\frac{(2+\text{x})^\frac{5}{2}-(\text{a}+2)^\frac{5}{2}}{(2+\text{x})-(\text{a}+2)}$
$=\frac{5}{2}(\text{a}+2)^{\frac{5}{2}-1}$
$=\frac{5}{2}(\text{a}+2)^{\frac{3}{2}}$
Hence, the required answre is $\frac{5}{2}(\text{a}+2)^{\frac{3}{2}}.$

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