Limit Example

Find the value of \lim_{n \to \infty}\left(1 - \frac{1}{n}\right)^{2n}

 

Answer

y =\lim\limits_{n \to \infty}\left(1-\frac{1}{n}\right)^{2n}

 

taking log on both sides we get:

\log y =\lim\limits_{n \to \infty} 2n \log \left(1-\frac{1}{n}\right)

=\lim\limits_{n \to \infty} \dfrac{\log \left(1-\frac{1}{n}\right)}{\left(\frac{1}{2n}\right)}

apply L' hospital rule:-

\log y=\lim\limits_{n \to \infty} \dfrac{\left(\dfrac{1}{1-\frac{1}{n}}\right).\frac{1}{n^{2}}}{\left(\dfrac{-1}{2n^{2}}\right)}

\log y={-2}

y=e^{-2}.

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