Question

If $\underset{x\to 0}{lim}(1+ax{)}^{\frac{b}{x}}=e^4$ where $a$ and $b$ are natural numbers, then

• A. $ab=8$
• B. $a=8,b=4$
• C. $ab=4$
• D. $a=4,b=8$

Answer 1

The correct option is C $ab=4$

$\underset{x\to 0}{lim}(1+ax{)}^{\frac{b}{x}}\to 1^{\infty }\text{form}$
$\underset{x\to 0}{lim}\frac{b(1+ax)}{x}=4\log e$
$\underset{x\to 0}{lim}\frac{b+abx}{x}=4$
Apply L'Hospital
$\Rightarrow ab=4$