Question

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

• A. $0$
• B. $1$
• C. $3$
• D. None of these

Answer 1

The correct option is B

$1$

Explanation for the correct option:

Find the value of $\underset{x\to 0}{\lim }\frac{\log \left(1+x^3\right)}{{\sin }^{3}x}$

Consider the given equation as

$I=\underset{x\to 0}{\lim }\frac{\log \left(1+x^3\right)}{{\sin }^{3}x}$

We know that

$$\begin{gathered} \underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\frac{\mathbf{log}\left(1+x\right)}{\mathbf{x}}\mathbf{=}\mathbf{1} \\ and\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\frac{\mathbf{sin}\mathbf{}\mathbf{x}}{\mathbf{x}}\mathbf{=}\mathbf{1} \end{gathered}$$

Then,

$$\begin{gathered} I=\underset{x\to 0}{\lim }\frac{\log \left(1+x^3\right)}{x^3\times {\displaystyle \frac{{\sin }^{3}x}{x^3}}} \\ \Rightarrow I=\frac{1}{1} \\ \Rightarrow I=1 \end{gathered}$$

Since,

$\underset{x\to 0}{\lim }\frac{\log \left(1+x^3\right)}{{\sin }^{3}x}=1$

Hence, the correct answer is option B