Measuring Infinities

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. $\frac{small\infty }{big\infty }$ = 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. $\frac{big\infty }{small\infty }$ = $\infty$. ${\lim }_{x\to -\infty }f(x)=\infty$