Question

$\underset{x\to a}{lim}\frac{(x+2{)}^{3/2}-(a+2{)}^{3/2}}{x-a}$

Answer 1

$\underset{x\to a}{lim}\frac{(x+2{)}^{3/2}-(a+2{)}^{3/2}}{x-a}$

$=\underset{x\to a}{lim}\frac{(x+2{)}^{3/2}-(a+2{)}^{3/2}}{(x+2)-(a+2)}$

Let $x+2=y$, $a+2=b$

$\therefore \underset{x+2\to a+2}{lim}\frac{(y{)}^{3/2}-(b{)}^{3/2}}{y-b}$

$[\because \underset{x\to a}{lim}\frac{x^n-a^n}{x-a}=n{a}^{n-1}]$

$=\frac{3}{2}(a+2{)}^{\frac{3}{2}-1}$

$=\frac{3}{2}(a+2{)}^{\frac{1}{2}}$.