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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