If f(x)=4x−13sin⁡(x/4)log⁡1+x2/3,x≠0k,x=0 is continuous at x = 0, then k =

A
$12 {(\log_{e} 4)}^{2}$
B
$96 (\log 2)^{3}$
C
${(\log_{e} 4)}^{3}$
D
none of these

Solution:

N Since $f (x)$ is continuous at $x = 0 .$ Therefore,

$\begin{aligned} lim_{x \to 0} f (x) = f (0) \\ \Rightarrow lim_{x \to 0} \frac{{(4^{x} - 1)}^{3}}{\sin \frac{x}{4} \log (1 + \frac{x^{2}}{3})} = k \\ \Rightarrow lim_{x \to 0} \frac{12 (\frac{4^{x} - 1^{3}}{x})}{(\frac{\sin \frac{x}{4}}{x / 4}) (\frac{\log (1 + x^{2} / 3)}{x^{2} / 3})} = k \end{aligned}$

$\Rightarrow 12 (\log 4)^{3} = k \Rightarrow k = 96 (\log 2)^{3}$

If $f (x) = \{\begin{array}{cc} \frac{{(4^{x} - 1)}^{3}}{\sin (x / 4) \log (1 + x^{2} / 3)}, & x \neq 0 \\ k & , x = 0 \end{array}$
is continuous at $x = 0,$ then $k =$

Solution:

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