Answer
$\lim\limits_{x \to +\infty} \frac{x^{100}}{e^{x}} =0$
Work Step by Step
$\lim\limits_{x \to +\infty} \frac{x^{100}}{e^{x}} =\lim\limits_{x \to +\infty} \frac{ \frac{d(x^{100})}{dx} }{ \frac{d(e^{x})}{dx}} $
$\lim\limits_{x \to +\infty} \frac{x^{100}}{e^{x}} =\lim\limits_{x \to +\infty} \frac{100x^{99}}{e^{x}} $
Applying L’Hôpital’s rule repeatedly
$\lim\limits_{x \to +\infty} \frac{x^{100}}{e^{x}} =\lim\limits_{x \to +\infty} \frac{100 \times 99 \times 98 \times ......1 x^{0}}{e^{x}} $
$\lim\limits_{x \to +\infty} \frac{x^{100}}{e^{x}} =\lim\limits_{x \to +\infty} \frac{100 \times 99 \times 98 \times ......1 x^{0}}{e^{x}} $
$\lim\limits_{x \to +\infty} \frac{x^{100}}{e^{x}} =\lim\limits_{x \to +\infty} \frac{100!}{e^{x}} $
$\lim\limits_{x \to +\infty} \frac{x^{100}}{e^{x}} =(100!)\lim\limits_{x \to +\infty} \frac{1}{e^{x}} $
$\lim\limits_{x \to +\infty} \frac{x^{100}}{e^{x}} =(100!)\lim\limits_{x \to +\infty} \times 0 $
$\lim\limits_{x \to +\infty} \frac{x^{100}}{e^{x}} =(100!) \times 0=0$
