Question

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

• A. $\frac{5}{3}(a+2{)}^{\frac{2}{3}}$
• B. $\frac{5}{3}(a+2{)}^{\frac{5}{3}}$
• C. $\frac{5}{3}{a}^{\frac{2}{3}}$
• D. $\frac{10}{3}{a}^{\frac{2}{3}}$

Answer 1

The correct option is A $\frac{5}{3}(a+2{)}^{\frac{2}{3}}$

${lim}_{x\to a}\frac{{(x+2)}^{\frac{5}{3}}-{(a+2)}^{\frac{5}{3}}}{(x-a)}=l$ (say) (is of the form $\left(\frac{0}{0}\right)$)

$l={lim}_{x\to a}\frac{\frac{5}{3}{[x+2]}^{\frac{2}{3}}\left(1\right)}{1}$ [L' Hospital Rule]

$l=\frac{5}{3}{[a+2]}^{\frac{2}{3}}$