Question

Evaluate the following limits :
$\underset{x\to a}{lim}\frac{{(x+2)}^{3/2}-{(a+2)}^{3/2}}{x-a}$

• A. $0$
• B. $\frac{3(a+2{)}^{\frac{1}{2}}}{2}$
• C. $1$
• D. $2$

Answer 1

The correct option is B $\frac{3(a+2{)}^{\frac{1}{2}}}{2}$

Given limit is of the form

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

Hence,

$\underset{x\to a}{lim}\frac{{(x+2)}^{3/2}-{(a+2)}^{3/2}}{(x+2)-(a+2)}=\underset{x\to a}{lim}\frac{(x+2{)}^{\frac{3}{2}}-(a+2{)}^{\frac{3}{2}}}{x-a}=\frac{3}{2}(a+2{)}^{\frac{1}{2}}$................[from $\left(1\right)$]