Question

If $a$ and $b$ are chosen randomly by throwing a pair of fair dice then the probability that $\underset{x \rightarrow 0}{\text{Lim}} \left(\frac{a^x+b^x}{2}\right)^{\frac{2}{x}}=6$ equals

• A. $\frac{4}{9}$
• B. $\frac{2}{9}$
• C. $\frac{3}{9}$
• D. $\frac{1}{9}$

Answer 1

Answer: (D)

$e ^{\text{Lim}_{x \rightarrow 0 x} \frac{2}{}\left(\frac{a^x-1+b^x-1}{2}\right)}=e^{\ln a b}=a b=6$
$(a, b)=(1,6)(6,1)(2,3)(3,2) $
$\Rightarrow P(E)=\frac{4}{36}=\frac{1}{9} $