lim-n-f-a-1-n-f-a-1-n-n-

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Question Number 370 by 123456 last updated on 25/Jan/15

$$\underset{n\to \mathrm{\infty }}{\lim }{\left[\frac{f(a+\frac{1}{n})}{f(a-\frac{1}{n})}\right]}^n$$

Answered by prakash jain last updated on 24/Dec/14

$$\underset{n\to \mathrm{\infty }}{\lim ln}y=\underset{n\to \mathrm{\infty }}{\lim }\frac{\ln f(a+\frac{1}{n})-\ln f(a-\frac{1}{n})}{1/n}$$$$=\underset{n\to \mathrm{\infty }}{\lim }\frac{\frac{f{}^{\prime }(a+\frac{1}{n})}{f(a+\frac{1}{n})}\cdot (-\frac{1}{n^2})-\frac{f{}^{\prime }(a-\frac{1}{n})}{f(a-\frac{1}{n})}\cdot (+\frac{1}{n^2})}{-\frac{1}{n^2}}$$$$=\underset{n\to \mathrm{\infty }}{\lim }\frac{(-\frac{1}{n^2})[\frac{f{}^{\prime }(a+\frac{1}{n})}{f(a+\frac{1}{n})}\cdot +\frac{f{}^{\prime }(a-\frac{1}{n})}{f(a-\frac{1}{n})}]}{-\frac{1}{n^2}}$$$$=\frac{2f{}^{\prime }\left(a\right)}{f\left(a\right)}$$$$\underset{n\to \mathrm{\infty }}{\lim }y=\exp \left(\frac{2f{}^{\prime }\left(a\right)}{f\left(a\right)}\right)$$

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