Q76 / 160 · AP EAMCET 2020 (17 Sep Shift 2) · Mathematics

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

  • Correct answer
    A
    6
  • B
    9
  • C
    81
  • D
    18

Solution

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 \(\therefore p+q=6\)

Hence, the correct answer is option: 'A'.

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