If $\lim _{x \rightarrow \infty}\left(1+\frac{p}{x}\right)^{q x}=e^9$ where $p, q \in \mathbf{N}$ then $p+q=$

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Aicharya · Accepted Answer

Given, $\lim _{x \rightarrow \infty}\left(1+\frac{p}{x}\right)^{q x}=e^9$, Let $\frac{1}{x}=y \Rightarrow$ When $x \rightarrow \infty \Rightarrow y \rightarrow 0$ $\therefore \lim _{y \rightarrow 0}(1+p y)^{q / y}=e^9$ $\Rightarrow \quad\left\{\lim _{y \rightarrow 0}(1+p y)^{1 / y}\right\}^q=e^9$ $\Rightarrow \quad\left(e^p\right)^q=e^9 \Rightarrow p q=9$ Let we take $p=3 \Rightarrow q=3$ and $\therefore p+q=6$

If \(\lim _{x \rightarrow \infty}\left(1+\frac{p}{x}\right)^{q x}=e^9\) where \(p, q \in \mathbf{N}\) then…

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