Finding Limits Approaching Infinity for Improper Integrals

[lelandsthename]

Homework Statement

Lim ((e$$^{2}$$)-($$\frac{t-3}{3}($$ * e^$$^{\frac{t}{3}})$$))
t -> $$-\infty$$

[sorry for the formatting, I tried my best! that is "The limit as t approaches negative infinity of e squared minus (t-3/3) e to the t/3)]"]

Homework Equations

I am solving improper integrals to find out if this is convergent or divergent and do not remember how to show on paper to solve for a limit.

The Attempt at a Solution

Well, if memory serves, the limit as we approach negative infinity is a horizontal asymptote, right? But I only remember those silly little coefficient tricks to find a horizontal asymptote. Otherwise, I'd just look on a calculator. Someone tried to explain it to me and said I should think of it as going towards positive infinity first and then negate it ... but even with positive infinity I don't no where to start!

Your detailed instructions/explanations would be awesome! I really want to understand how to do this by hand.

 

[HallsofIvy]

$$\left(e^2- \frac{t-3}{3}\right)e^{-t/3}$$
or
$$e^2- \frac{t-3}{3}e^{-t/3}$$?

In either case it looks pretty straight forward to me: e^-x, times any power of t, goes to 0 as x goes to infinity.