limx→0 4x−13sin⁡x4loge⁡1+x23= - Sri Chaitanya Infinity Learn Best Online Courses for NCERT Solutions, CBSE, ICSE, JEE, NEET, Olympiad and Class 6 to 12

A
${(\log_{e} 4)}^{3}$
B
$\log_{e} 4$
C
$12 {(\log_{e} 4)}^{3}$
D
$5 {(\log_{e} 4)}^{3}$

Solution:

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

Divide with x³ and simplify

$= \frac{{(lim_{x \to 0} \frac{4^{x} - 1}{x})}^{3}}{lim_{x \to 0} \frac{\sin (\frac{x}{4})}{\frac{x}{4} . 4} lim_{x \to 0} \log_{e} (\frac{(1 + \frac{x^{2}}{3})}{\frac{x^{2}}{3} . 3})} = 12 {(\log_{e} 4)}^{3}$

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

Solution:

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