Question

The value of $a$ for which $\underset{x\to 0}{lim}\frac{(e^x-1{)}^2}{\sin \frac{x}{a}\log (1+\frac{x}{2})}=8$, is

• A. $1$
• B. $4$
• C. $\frac{5}{4}$
• D. $\frac{1}{4}$

Answer 1

The correct option is B $4$

$\underset{x\to 0}{lim}\frac{(e^x-1{)}^2}{\sin \frac{x}{a}\log (1+\frac{x}{2})}=8$
$\underset{x\to 0}{lim}\frac{{\left(\frac{e^x-1}{x}\right)}^2}{\frac{\sin \frac{x}{a}}{\frac{ax}{a}}\frac{\log (1+\frac{x}{2})\frac{1}{2}}{\frac{x}{2}}}=8$
$\frac{1}{\frac{1}{a}\frac{1}{2}}=8$
$\Rightarrow a=4$