How do I find #lim_(x>oo)(3sin(x))/e^x# using power series?
2 Answers
To be honest, I would not use power series on this one since this is a perfect problem to demonstrate the application of Squeeze Theorem. Here is how:
We know
Since
and
we conclude that
by Squeeze Theorem.
I hope that this was helpful.
The power series in question are:

#sin(x)=sum_(n=0)^oo(1)^nx^(2n+1)/((2n+1)!)=xx^3/(3!)+x^5/(5!)x^7/(7!)+...# 
#e^x=sum_(n=0)^oox^n/(n!)=1+x+x^2/(2!)+x^3/(3!)+x^4/(4!)+...#
So:
In the numerator, we see that