Question

Evaluate the following limits:

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

Answer 1

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

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

Let y =x+2 and b =a+2

When $x\Rightarrow a,$

Then $x+2\to a+2$

$\Rightarrow y\to b$

$={lim}_{y\to b}\frac{y^{\frac{5}{2}-b^{\frac{5}{2}}}}{y-b},$

$=\frac{5}{2}{b}^{\frac{5}{2}-1}$

[Using formula ${lim}_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}]$

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

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