🍈 Zettelkasten

Problem

Say we have a $\frac{x ^{2}}{e ^{x}}$ How do we know what it approaches, when $x \to \infty$ ?

Measuring Infinities

You must find when one function is bigger than another function. Equate the 2 functions together, then find critical values. $x^{2} = e^{x}$ $x^{2} - e^{x} = 0$ This will =be the function for finding if $x^{2}$ is bigger than $e^{x}$ . $x = - 0.703$ Now check intervals

Test	$x < - 0.703$	$x > - 0.703$
$x^{2} - e^{x}$	+	-
So, $x^{2} > e^{x}$ when $x < - 0.703$
$x^{2} < e^{x}$ when $x > - 0.703$

$lim_{x \to \infty} f (x)$ will follow $x > - 0.703$ so, $e^{x}$ will be bigger. $e^{x}$ is the bigger infinity. $\frac{s ma ll \infty}{bi g \infty}$ = 0. $lim_{x \to \infty} f (x) = 0$

$lim_{x \to - \infty} f (x)$ will follow $x < - 0.703$ so, $x^{2}$ will be bigger. $\frac{bi g \infty}{s ma ll \infty}$ = $\infty$ . $lim_{x \to - \infty} f (x) = \infty$