x 0lim(((1- ^2(3x) ^3(4x))( ^3(4x)( _e(2x+1))^5)) is equal to _____________.

Question

$\underset{x	o 0}{lim}((\frac{(1-{\cos }^{2}(3x)}{{\cos }^{3}(4x)})(\frac{{\sin }^{3}(4x)}{{({\log }_{e}(2x+1))}^{5}}))$ is equal to \_\_\_\_\_\_\_\_\_\_\_\_\_.

Accepted Answer

$\begin{gathered} \underset{x	o 0}{lim}[(\frac{1-{\cos }^{2}(3x)}{{\cos }^{3}(4x)})(\frac{{\sin }^{3}(4x)}{{({\log }_{e}(2x+1))}^{5}})] \ =\underset{x	o 0}{lim}[\frac{1-{\cos }^{2}(3x)}{9{x}^{2}}	imes \frac{9{x}^{2}}{{\cos }^{3}(4x)}]	imes \frac{\frac{{\sin }^{3}4x}{(4x{)}^{3}}	imes 64{x}^{3}}{{[\frac{{\log }_{e}(2x+1)}{2x}]}^{5}	imes (2x{)}^{5}} \ =[\frac{1	imes 9	imes 1}{(1)}]	imes [\frac{1	imes 64}{1	imes 32}] \ =9	imes 2=18 \end{gathered}$

Solutions · 1