How to Simplify a Complicated Limit

[Bipolarity]

Homework Statement

Been trying to evaluate this rather annoying limit for the past few minutes.

\lim_{t→∞}\frac{t-\frac{1}{ln(t)}}{(ln(t))e^{at-\frac{t}{ln(t))}}}

Homework Equations

The Attempt at a Solution

I tried L'Hopital's rule but it seems to become only more messy. I was curious if there was any "quick" way to this problem? It's been a while since I took calculus, and I'm using this as part of a Laplace transform I'm trying to do.

BiP

 

[Dick]

Homework Statement

Been trying to evaluate this rather annoying limit for the past few minutes.

\lim_{t→∞}\frac{t-\frac{1}{ln(t)}}{(ln(t))e^{at-\frac{t}{ln(t))}}}

Homework Equations

The Attempt at a Solution

I tried L'Hopital's rule but it seems to become only more messy. I was curious if there was any "quick" way to this problem? It's been a while since I took calculus, and I'm using this as part of a Laplace transform I'm trying to do.

BiP

Try and simplify it first. Drop terms that aren't important. In the numerator 1/ln(t) approaches 0, t approaches infinity. So you can drop the 1/ln(t) without affecting the limit. In the denominator, if a>0, then t/ln(t) in the exponent goes to infinity, but it is dominated by at going to infinity faster (to see this check that the ratio (t/ln(t))/(at) goes to zero), so drop that. That should give you something easier to do l'Hopital from. As you do l'Hopital, keep checking for terms that grow more slowly than others and keep simplifying.