If $f(x) = \left\{ \begin{array}{r} \frac{\left( 4^{x} - 1 ight)^{3}}{\sin{\left( \frac{x}{a} ight)\log\left( 1 + \frac{x^{2}}{3} ight)}},\ x
eq 0 \ 9\left( \log 4 ight)^{3},\ x = 0 \end{array} ight.\$ is continuous function at $x = 0$, then the value of $a$ is equal to

Question

If $f(x) = \left\{ \begin{array}{r} \frac{\left( 4^{x} - 1 ight)^{3}}{\sin{\left( \frac{x}{a} ight)\log\left( 1 + \frac{x^{2}}{3} ight)}},\ x 
eq 0 \ 9\left( \log 4 ight)^{3},\ x = 0 \end{array} ight.\$ is continuous function at $x = 0$, then the value of $a$ is equal to

Dr. Abubacker · Accepted Answer

(a)
$\lim_{x ightarrow 0}f(x) = \lim_{x ightarrow 0}\left( \frac{4^{x} - 1}{x} ight)^{3} 	imes \frac{\left( \frac{x}{a} ight)}{\sin\left( \frac{x}{a} ight)}.\frac{ax^{2}}{\log\left( 1 + \frac{1}{3}x^{2} ight)}$
$= \left( \log 4 ight)^{3}.1.a\ \lim_{x ightarrow 0}\left( \frac{x^{2}}{\frac{1}{3}x^{2} - \frac{1}{18}x^{4} + ...} ight)$
$= 3a\ \left( \log 4 ight)^{3}$
$\because\ \ \ \ \lim_{x ightarrow 0}\ f(x) = f(0)$
$\Rightarrow \ \ 3a\ \left( \log 4 ight)^{3} = 9\left( \log 4 ight)^{3}$
$\Rightarrow \ \ \ \ \ a = 3$

Q. If $f(x) = \left\{ \begin{array}{r} \frac{\left( 4^{x} - 1 \right)^{3}}{\sin{\left( \frac{x}{a} \right)\log\left( 1 + \frac{x^{2}}{3} \right)}},\ x \neq 0 \\ 9\left( \log 4 \right)^{3},\ x = 0 \end{array} \right.\$ is continuous function at $x = 0$, then the value of $a$ is equal to

Explanation

Reviewed by Dr. Abubacker

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