The value of limx→0 4x−13sin⁡x24log⁡(1+3x), is

A
$\frac{4}{3} (\ln 4)^{2}$
B
$\frac{4}{3} (\ln 4)^{3}$
C
$\frac{3}{2} (\ln 4)^{2}$
D
$\frac{3}{2} (\ln 4)^{5}$

Solution:

We have,

$lim_{x \to 0} \frac{{(4^{x} - 1)}^{3}}{\sin \frac{x^{2}}{4} \log (1 + 3 x)}$

$= lim_{x \to 0} \frac{{(\frac{4^{x} - 1}{x})}^{3}}{\frac{1}{4} \{\frac{\sin (x^{2} / 4)}{\frac{x^{2}}{4}}\} \times \frac{\log (1 + 3 x)}{3 x} \times 3} = \frac{4}{3} \log_{e} 4^{3}$

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