Question

$\underset{x\to 0}{lim}\frac{\sqrt[3]{31+3x}-1-x}{(1+x{)}^{101}-1-101x}$ has the value equal to

• A. $-\frac{3}{5050}$
• B. $-\frac{1}{5050}$
• C. $\frac{1}{5051}$
• D. $\frac{1}{4950}$

Answer 1

The correct option is B $-\frac{1}{5050}$

Applying L'Hospital's Rule,
$={lim}_{x\to 0}\frac{\frac{1}{3}3(1+3x{)}^{\frac{-2}{3}}-1}{101(1+x{)}^{100}-101}$
$={lim}_{x\to 0}\frac{\frac{-2}{1}(1+3x{)}^{\frac{-5}{3}}}{101.100(1+x{)}^{99}}$
$=\frac{-2}{101.100}$$=-\frac{1}{5050}$