f-n-x-1-x-n-nx-n-N-x-R-x-0-lim-n-f-n-x-

Question Number 585 by 123456 last updated on 02/Feb/15

f_{n} (x) = {(1 + \frac{x}{n})}^{n x}, n \in N^{*}, x \in R, x ⩾ 0

\lim_{n \to + \infty} f_{n} (x) =

Answered by prakash jain last updated on 02/Feb/15

y = {(1 + \frac{x}{n})}^{n x}

\ln y = n x \ln (1 + \frac{x}{n}) = \frac{x \ln (1 + \frac{x}{n})}{\frac{1}{n}}

\underset{n \to \infty}{\lim l n} y = \lim_{n \to \infty} x \cdot \frac{\frac{- \frac{x}{n^{2}}}{1 + \frac{x}{n}}}{- \frac{1}{n^{2}}} = \frac{x^{2}}{1 + \frac{x}{n}} = x^{2}

\lim_{n \to \infty} f_{n} (x) = e^{x^{2}}