evaluate-lim-x-pi-api-x-x-

Question Number 33865 by 33 last updated on 26/Apr/18

e v a l u a t e

l i \underset{x \to \infty}{m} π \frac{{(a π)}^{x}}{x!}

Commented by abdo imad last updated on 26/Apr/18

w e h a v e f o r x \in V (+ \infty) x! \sim x^{x} e^{- x} \sqrt{2 π x} (s t i r l i n g f o r m u l a g e n e r a l i s e d)

\Rightarrow A (x) = \frac{π {(a π)}^{x}}{x!} \sim \frac{π {(a π)}^{x}}{x^{x} e^{- x} \sqrt{2 π x}} = π e^{x} {(a x)}^{x} x^{- x - \frac{1}{2}} . \frac{1}{\sqrt{2 π}}

= \frac{\sqrt{π}}{2} e^{x} a^{x} x^{- \frac{1}{2}} w e m u s t h a v e a > 0 a n d a \neq \frac{1}{π} \Rightarrow_{}

A (x) \sim \frac{\sqrt{π}}{2} e^{x l n (a e)} e^{- \frac{1}{2} l n (x)} = \frac{\sqrt{π}}{2} e^{x l n a + x - \frac{1}{2} l n (x)} \Rightarrow

A (x) \sim \frac{\sqrt{π}}{2} e^{x (l n a + 1 - \frac{1}{2} \frac{l n x}{x})} \to + \infty (x \to + \infty)

Commented by abdo imad last updated on 26/Apr/18

l i m A (x) = + \infty i f l n (a) + 1 > 0 \Leftrightarrow a > \frac{1}{e}

i f 0 < a < e l n (a) + 1 < 0 \Rightarrow {l i m}_{x \to + \infty} A (x) = 0