Question

$\underset{x\to 0}{lim}\frac{(1+x{)}^2-(1-x{)}^2}{(1+x{)}^3-(1-x{)}^3}$=

• A. 1
• B. $\frac{1}{2}$
• C. $\frac{2}{3}$
• D. $\frac{3}{2}$

Answer 1

The correct option is C $\frac{2}{3}$

We know that, $\underset{x\to 0}{lim}(1+x{)}^n\approx 1+nx$
Thus,
$\underset{x\to 0}{lim}\frac{(1+x{)}^2-(1-x{)}^2}{(1+x{)}^3-(1-x{)}^3}$
$=\underset{x\to 0}{lim}\frac{(1+2x)-(1-2x)}{(1+3x)-(1-3x)}=\underset{x\to 0}{lim}\frac{1+2x-1+2x}{1+3x-1+3x}={lim}_{x\to 0}\frac{4x}{6x}=\frac{2}{3}$