# Math Help - Finding limits of an integral using L'hopital rule

1. ## Finding limits of an integral using L'hopital rule

Here is the solution for the given question attached as a picture since I'm still not so used to using Latex yet...

What confuses me is how they go from the integral to the next part. I know for L'hopital's rule you take the function as a quotient p(t)/q(t) then you find their respective derivatives. However when the top part is a definite integral how do you approach it?

2. ## Re: Finding limits of an integral using L'hopital rule

let $u$ be a function of $t$ and $a$ a constant; the fundamental theorem of calculus states ...

$\frac{d}{dt} \int_a^u f(v) \, dv = f(u) \cdot \frac{du}{dt}$

so, the derivative of the numerator is ...

$\frac{d}{dt} \left(-\int_{\infty}^{\log{t}} e^{-e^v} \, dv \right) = -e^{-e^{\log{t}}} \cdot \frac{1}{t} = -e^{-t} \cdot \frac{1}{t}$

3. ## Re: Finding limits of an integral using L'hopital rule

wow thanks, didnt think of that. Was trying to solve it with leibniz rule etc which just overcomplicated things. So the negative sign is added because we reversed the domain which the integral is defined over?

Just gave you your 100th thanks xD

4. ## Re: Finding limits of an integral using L'hopital rule

Originally Posted by holaboo
So the negative sign is added because we reversed the domain which the integral is defined over?
correct ... just switched the limits of integration.