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  4. displaystyle lim x arrow 0 ( ln (2- cos 2 x)/ ln 2( sin 3 x+1)) is equal to

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Q. $\displaystyle\lim _{x \rightarrow 0} \frac{\ln (2-\cos 2 x)}{\ln ^{2}(\sin 3 x+1)}$ is equal to

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 0} \frac{\ln (2-\cos 2 x)}{\ln ^{2}(\sin 3 x+1)}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{\ln \{1+(1-\cos 2 x)\}}{\ln ^{2}(1+\sin 3 x)}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{(\sin 3 x)^{2}}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{2 x^{2}}{(3 x)^{2}}=\frac{2}{9}$