Evaluate the following limit: _ x 0 |1+x^3 | ^3x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\log\big|1+\text{x}^3\big|}{\sin^3\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\log\big|1+\text{x}^3\big|}{\sin^3\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\log\big|1+\text{x}^3\big|}{\sin^3\text{x}}\times\frac{1}{\lim\limits_{\text{x}\rightarrow0}\big(\frac{\sin\text{x}}{\text{x}}\big)^3}$
$=1\times1$
$=1$

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