# L'Hospital's Rule

1. Feb 22, 2013

### maceng7

Hey I'm doing indeterminate products and using l'hospital's rule and does anyone have a strategy that works well in determining which function you put as your denominator? Obviously there isn't one strategy that works 100% but I'm just wondering if anyone knows a good way of determining this? Thanks.

2. Feb 22, 2013

### SammyS

Staff Emeritus
Re: L'Hospital's Rule Indeterminate Products

You're right, there is no general rule.

Whatever gives better cancellation of offending factors is often a good choice.

3. Feb 22, 2013

### maceng7

Re: L'Hospital's Rule Indeterminate Products

What's a good way to start then? Putting the function whose derivative is simpler on the bottom or top or does it even matter?

4. Feb 22, 2013

### Bashyboy

Why don't you provide us with an problem that you are having difficulty with. As you mentioned, there is no one way of getting your indeterminate form fit 0/0 or infinty/infinity. So, giving us an example may allow us to help you.

5. Feb 22, 2013

### SammyS

Staff Emeritus
Re: L'Hospital's Rule Indeterminate Products

It's not always that simple, (Pardon the joke.) although that does often work.

I have seen cases where the derivative of top or bottom gets more complicated than the original, but the result affords just the right amount of cancelling with the other derivative so that the overall result is nicely behaved.
--- no example off the top of my head.​

6. Feb 22, 2013

### maceng7

Alright I think I did this one correctly

lim x--> 1+ ln(x) * tan(∏x/2)

I found this limit by putting ln(x) in the numerator and tan(∏x/2) in the denominator. I used l'hospital's rule once and found it to be 1/∞ so my answer is 0. Is this correct?

7. Feb 22, 2013

### Yosty22

You can only use L'Hospital's Rule when it is 0/0 or infinity/infinity. If it is 0/infinity, you have to do some algebra to rewrite it in a form that is 0/0 or infinity/infinity.

8. Feb 22, 2013

### SammyS

Staff Emeritus

9. Feb 22, 2013

### Bashyboy

What's the statement of the original problem? Is it $lim_{x \rightarrow 1^+}~ ln(x) \cdot \tan( \frac{\pi x}{2})$? I ask, because you can't just arbitrarily chose which function you want to put in the numerator and which one will go in the denominator. You have to use algebraic manipulation to put it in the form you want.

10. Feb 22, 2013

### Dick

No, it's not right. How exactly did you do that? You don't just 'put something into the denominator'. a*b is not equal to a/b, it is equal to a/(1/b).

11. Feb 22, 2013

### Bashyboy

It appears we are all giving the same advice. @maceng7 is what we are saying making any sense?

Let a = ln(x) and b = tan(pix/2). As dick mentioned, a*b is entirely different from a/b. By putting the b in the denominator, you've created a new function, entirely different from the original; and when you evaluate the limit of a/b, the result you get tells you nothing of the limit for a*b. This is true, because a*b and a/b are different functions.

Last edited: Feb 22, 2013
12. Feb 22, 2013

### Ray Vickson

NO: if it is 0/∞ you are done: the answer = 0 and there is no need for l'Hospital's rule.

13. Feb 23, 2013

### SammyS

Staff Emeritus
But as Dick point's out, the limit is not zero so ...

Let's wait to hear back from OP .