If lim _{x rightarrow infty}left(1+frac{p}{x} | vedclass

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Question

(A) Given,$\lim _{x \rightarrow \infty}\left(1+\frac{p}{x}\right)^{q x}=e^9$. Using the standard limit formula $\lim _{x \rightarrow \infty}(1+\frac{a

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(A) Given,$\lim _{x \rightarrow \infty}\left(1+\frac{p}{x}\right)^{q x}=e^9$. Using the standard limit formula $\lim _{x \rightarrow \infty}(1+\frac{a}{x})^x = e^a$,we have: $\lim _{x \rightarrow \infty}\left(1+\frac{p}{x}\right)^{q x} = \left[ \lim _{x \rightarrow \infty} \left(1+\frac{p}{x}\right)^{x} \right]^{q} = (e^p)^q = e^{pq}$. Comparing this with the given expression $e^9$,we get $pq = 9$. Since $p, q \in \mathbb{N}$ (natural numbers),the possible pairs $(p, q)$ are $(1, 9), (3, 3), (9, 1)$. In these cases,$p+q$ can be $1+9=10$ or $3+3=6$. Given the options,the correct value is $6$.

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

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