Question

${lim}_{x\to a}\frac{(x+2{)}^{5/3}-(a+2{)}^{5/3}}{x-a}$

Answer 1

$\underset{x\to a}{lim}\frac{{(x+2)}^{5/3}-{(a+2)}^{5/3}}{x-a}$ (putting the limits we see that it is in $\frac{0}{0}$ form)

So we use $L^{\prime }$ Hospital's rule

$=\underset{x\to a}{lim}\frac{\frac{5}{3}{(x+2)}^{2/3}}{1}$

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