Question $2.$ Asymptotics

Another common approximation in applied math and engineering is to use only the leading order terms for functions that grow without bound. In this exercise we will recall how to use the $$big$-$O$$ notation to denote such an

approximation, and write some simple MATLAB code that can help us confirm when the definition can be applied.

$($a$)$ Consider the functions

f$1($x$)=$ e

x

$,$

and

f$2($x$)=$ e

$($x

$2$

$)$

$.$

Is it true that f$1($x$)$ in O$($f$2($x$))$ or that f$2($x$)$ in O$($f$1($x$))$ as x $->\backslash$infty $?$ Justify your answer by taking a limit$.$

$($b$)$ When we write the expression $$limn$->\backslash$infty f$($n$)=$ L$,$ this really means that $$we can ensure that f$($n$)$ will be

as close to L as we like, provided we take n sufficiently large$.$ Write MATLAB code that takes as input a

symbolic function f$,$ a limit L$,$ and an error threshold err, and returns the smallest positive integer n for which

$|$f$($n$)$ L$|<$ err. Examining the limit you considered in $2($a$),$ use your code to produce the smallest integer n

for which the ratio of functions you considered is within $106$ of its limiting value.