Question

Evaluate : \( \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{ pn + q } \)

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Please answer as soon as possible

Answer 1

\(\lim\limits_{n\to\infty}(1+\frac1n)^{pn+q}\) (1^\(\infty\) type)

= Exp{\(\lim\limits_{n\to\infty}(1+\frac1n-1)^{pn+q}\)}

 = Exp {\(\lim\limits_{n\to\infty}\frac{pn+q}n\)}

 = Exp{\(\lim\limits_{n\to\infty}(p+\frac qn)\)}

= Exp {P}

= Exp^P 

∴ \(\lim\limits_{n\to\infty}(1+\frac 1n)^{pn+q}\) = e^P