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 $\mathrm{f}$ is continuous at $x=0$, then the value of $a$ is equal to

Question

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 $\mathrm{f}$ is continuous at $x=0$, then the value of $a$ is equal to, 1, -1, 2, 3

MARKS App · Accepted Answer

$\begin{aligned} & \operatorname{Lt}_{x \rightarrow 0} \frac{\left(e^x-1\right)^2}{\sin \left(\frac{x}{a}\right) \log \left(1+\frac{x}{4}\right)} \ &=\operatorname{Lt}_{x \rightarrow 0} \frac{\frac{\left(e^x-1\right)^2}{x} \cdot x^2}{\frac{x}{a} \cdot \frac{\sin \left(\frac{x}{a}\right)}{\left(\frac{x}{a}\right)} \cdot \frac{\log \left(1+\frac{x}{4}\right)}{\frac{x}{4}} \cdot \frac{x}{4}} \Rightarrow 4 a=12 \ & \Rightarrow a=3 \end{aligned}$

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