Question

Evaluate: $\underset{x\to 0}{lim}\frac{e^{x^3}-(1+x^3)}{{\sin }^6\left(2x\right)}$

• A. $\frac{1}{128}$
• B. $\frac{2}{127}$
• C. $\frac{1}{126}$
• D. $\frac{1}{125}$

Answer 1

The correct option is A $\frac{1}{128}$

$\underset{x\to 0}{lim}\frac{e^{x^3}-(1+x^3)}{{\sin }^6\left(2x\right)}$

$=\underset{x\to 0}{lim}\frac{(1+x^3+\frac{x^6}{2!})\left[\text{higher power neglected}\right]-(1+x^3)}{{\sin }^6\left(2x\right)}$

$=\frac{1}{2}\underset{x\to 0}{lim}\frac{x^6}{{\sin }^6\left(2x\right)}$

$=\frac{1}{2\cdot 2^6}{lim}_{x\to 0}{\left(\frac{2x}{\sin \left(2x\right)}\right)}^6$

$=\frac{1}{2^7}$

$=\frac{1}{128}$