Let $f(x) = \frac{\left( e^{x} - 1 ight)^{2}}{\sin{\left( \frac{x}{a} ight)\log\left( 1 + \frac{x}{4} ight)}}$ for $x
eq 0$ and$f(0) = 12$. If $f$ is continuous at $x = 0$, then the value of $a$ is equal to

Question

Let $f(x) = \frac{\left( e^{x} - 1 ight)^{2}}{\sin{\left( \frac{x}{a} ight)\log\left( 1 + \frac{x}{4} ight)}}$ for $x 
eq 0$ and$f(0) = 12$. If $f$ is continuous at $x = 0$, then the value of $a$ is equal to

Dr. Abubacker · Accepted Answer

(d)
$f(0) = \lim_{x \rightarrow 0}\frac{\left( e^{x} - 1 \right)^{2}}{\sin{\left( \frac{x}{a} \right)\log\left( 1 + \frac{x}{4} \right)}}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \lim_{x \rightarrow 0}\ \left( \frac{e^{x} - 1}{x} \right)^{2}.\ \frac{\frac{x}{a}.a}{\sin\frac{x}{a}}.\frac{\frac{x}{4}.4}{\log\left( 1 + \frac{x}{4} \right)} = 12$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ 1^{2}.a.4 = 12$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ a = 3$

Q. Let $f(x) = \frac{\left( e^{x} - 1 \right)^{2}}{\sin{\left( \frac{x}{a} \right)\log\left( 1 + \frac{x}{4} \right)}}$ for $x \neq 0$ and$f(0) = 12$. If $f$ is continuous at $x = 0$, then the value of $a$ is equal to

Explanation

Reviewed by Dr. Abubacker

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