Question

limit/l'hopital question
I'm struggling to figure out how to answer this without stepping through the math 8 times.

lim x-> inf. e^x/((x^2+3x+5)^8)

Answer 1

\[e ^{x}\div(x ^{2}+3x+5)^{8}\]

Answer 2

if u want just answer, you may use the below fact

exponential function overtakes ANY degree polynomial, in the extremes

Answer 3

I wrote down something similar but I need to evaluate the limit

Answer 4

It wouldnt be practical to write out all the steps. But if you had to do this via l'hopital's rule, you could just use a little bit of logic. The derivative of the numerator, e^x, will always be e^x. The denominator is just a polynomial, though. EVENTUALLY, you can take the derivative of it so many times that it will all eventually reduce down to a constant. If you multiplied it all out, all the way all 8 powers, you'd end up with some sort of polynomial degree 16. Well, you would take the derivative some 17 times and get a constant. So all you need to realize is that e^x stays in the numerator forever, the denominator eventually goes away, and you plug in infinity for x. So e^(infinity) = infinity.

Answer 5

Thanks a bunch. The logic flows far more clearly now.

Answer 6

Yeah, no problem : )

Answer 7

A solution and plot from Mathematica is attached.