If $a$ and $b$ are chosen randomly from the set consisting of numbers $1,2,3,4,5,6$ with replacement

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If $a$ and $b$ are chosen randomly from the set consisting of numbers $1,2,3,4,5,6$ with replacement.

Then the probability that $\lim_{x \rightarrow 0}\mspace{2mu}\left( \frac{a^{x} + b^{x}}{2} \right)^{2/x} = 6$ is

Accepted Answer

Sol. [Key Concept: $a,b$ from the set with $\log_{a}\ b$ an integer: count valid pairs.]$$\lim_{x ightarrow 0}\mspace{2mu}\left( \frac{a^{x} + b^{x}}{2} ight)^{2x} = e^{\lim_{x ightarrow 0}\mspace{2mu}\left( a^{x} + b^{x} - 1 ight)^{2}}\ ^{x} = e^{lnab} = ab$$$$\begin{matrix} \Rightarrow \ ab & \ = 6 \ (a,b) & \ = (1,6),(6,1),(2,3),(3,2) \ P & \ = \frac{4}{36} = \frac{1}{9} \end{matrix}$$

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