limx→∞ xnex=0,(n integer), for - Infinity Learn by Sri Chaitanya

If $n$ is a negative integer, then $n = - m,$ where $m \in N .$

$∴ lim_{x \to \infty} \frac{x^{n}}{e^{x}} = lim_{x \to \infty} \frac{x^{- m}}{e^{x}} = lim_{x \to \infty} \frac{1}{x^{m} e^{x}} = 0$

If $n = 0,$ then

If $n \in N,$ then,

$lim_{x \to \infty} \frac{x^{n}}{e^{x}} = lim_{x \to \infty} \frac{n!}{e^{x}} = 0$ [By L' Hospital's Rule]

Hence, $lim_{x \to \infty} \frac{x^{n}}{e^{x}} = 0$ for all values of $n$ .

$lim_{x \to \infty} \frac{x^{n}}{e^{x}} = 0, (n$ integer), for