Question

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

Answer 1

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

$\frac{0}{0}$ form, applying L'Hospital rule, we get

${lim}_{x\to a}\frac{\frac{5}{2}{(x+2)}^{5/2-1}}{1-0}$

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

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