Finding the limit of (1 - 5x)^(3/x) as x->0

[duki]

Homework Statement

lim of x->0 from above of (1-5x)^\frac{3}{x}

The Attempt at a Solution

I get to y=(1-5x)^\frac{3}{x} but I'm stuck here.

 

[Dick]

What's the limit x->infinity of (1+1/x)^x? I'm guessing you know that. Can you rearrange your limit into something in that form. Try a change of variable like 1/u=-5x.

 

[duki]

Ok I get to this part:

ln y = \frac{3}{x} ln(1-5x)
I know the next step should be
\frac{\frac{3}{1-5x}\frac{d}{dx}(1-5x)}{1}

but why?

 

[Dick]

You are trying to apply l'Hopital's theorem to the log, aren't you? I didn't recognize it. Ok. The log of y is (3/x)*ln(1-5x). Write that as 3*ln(1-5x)/x. That has a 0/0 form. So you can use l'Hopital. It has the form f(x)/g(x) where f(x) and g(x) go to zero. What does l'Hopital say?

 

[duki]

You're right, I don't =(

The limit you gave me, is it infinity?

 

[Dick]

You're right, I don't =(

The limit you gave me, is it infinity?

No, it's e. But that's ok. If you can use l'Hopital it's easier than that. I edited my post a lot after I realized that. Reread it, ok?

 

[duki]

Alright, thanks.

I got \frac{\frac{3(-5)}{1-5x}}{x} so far. Is that right?

 

[Dick]

According to l'Hopital, the candidate for the limit is f'(x)/g'(x), right? You took the derivative of the numerator. Why didn't you take the derivative of the denominator?

 

[duki]

Oh, ok. Oops.

so (1-5x) * (-15) ?

 

[Dick]

Right. Now let x->0. What do you get? And remember this is ln(y). So what is y?

 

[duki]

ahhh OK. So does y = e^{-15} ?

 

[Dick]

Yes. Now, if you want, tell me as an exercise why the limit I originally asked about, lim x->infinity (1+1/x)^x=e. It's the same thing really. Would be just for fun. If you get that you have it.

 

[duki]

hmm

I get to ln y = x ln (1+\frac{1}{x}) and I'm stuck. Is that 0/0? If so, how? Or is that infinity over infinity?

 

[Dick]

It's infinity*zero. Change it to ln(1+1/x)/(1/x). Now it's 0/0. Now you can apply l'Hopital. Find f'(x)/g'(x) and let x->infinity. It's easy. You're almost there.

 

[duki]

Ok. So when you differentiate you get -1 / -1. So then ln y = 1 and y = e^1 = e ?

 

[Dick]

Yes. You win! It's e. Remember the trick that if you see a limit like 0*infinity you can change it into infinity/infinity or 0/0 by moving a term into the numerator or denominator, whichever makes the derivatives easier. Well done.

 

[duki]

Awesome! Thanks.
Don't happen to have any more practice problems handy do you? :)

I have a test tomorrow on l'Hopital, and integration by simple sub, trig sub, partial fractions, and parts. I've been doing the same problems so many times, I'm worried I've just remembered the problems and not the techniques.

 

[Dick]

Nah. Don't have a big pool of practice problems. I'm sure you have plenty. Good luck. You did pretty well tonight once you got your wits about you. Get some sleep.

 

[duki]

Ok. Thanks again for your help!