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Solve the following

Question:

$\lim _{x \rightarrow a} \frac{(2+x)^{\frac{5}{2}}-(a+2)^{\frac{5}{2}}}{x-a}$

Solution:

Given $\lim _{x \rightarrow a} \frac{(2+x)^{\frac{5}{2}}-(a+2)^{\frac{5}{2}}}{x-a}$

$\Rightarrow \lim _{x \rightarrow a} \frac{(2+x)^{\frac{5}{2}}-(a+2)^{\frac{5}{2}}}{x-a}$

Now by adding and subtracting 2 to denominator for further simplification we get

$=\lim _{x \rightarrow a} \frac{(2+x)^{\frac{5}{2}}-(a+2)^{\frac{5}{2}}}{(x+2)-(a+2)}$

Now we have $\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}=n a^{n-1}$

$\Rightarrow \lim _{x \rightarrow a} \frac{(2+x)^{\frac{5}{2}}-(a+2)^{\frac{5}{2}}}{(x+2)-(a+2)}$

By using the above formula we get'

$=\frac{5}{2}(a+2)^{\frac{5}{2}-1}$

Simplifying and applying the limits we get

$=\frac{5}{2}(a+2)^{\frac{3}{2}}$

$\Rightarrow \lim _{x \rightarrow a} \frac{(2+x)^{\frac{5}{2}}-(a+2)^{\frac{5}{2}}}{x-a}=\frac{5}{2}(a+2)^{\frac{3}{2}}$

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