# L'Hopital's rule

1. Jul 6, 2010

### RockPaper

1. The problem statement, all variables and given/known data
lim as x -> 0 sin(e^(x^2)-1)/e^(cosx)-e

I saw this on the net and tried to solve it. I'm wondering if I'm correct by any means or did I mess up somewhere along the line.

2. Relevant equations

3. The attempt at a solution
http://img291.imageshack.us/img291/1174/hopitalsrule.jpg [Broken]

Last edited by a moderator: May 4, 2017
2. Jul 6, 2010

### Bohrok

The second differentiation of the numerator isn't correct. It shouldn't be just one term but two.
(f*g)' = fg' + gf'

3. Jul 6, 2010

### njama

You did it well.

Regards.

4. Jul 6, 2010

### RockPaper

I think you're right Bohrok (because I have a tendency of messing up).

@Bohrok: Do you mean the second differentiation should be

-sin(ex2-1)2xex2+(2ex2+4x2ex2)cos(ex2-1)/-cos(x)ecosxsin2x*ecosx

also, would the negative signs go away since we have both in the numerator and denominator? could we reduce the answer to 0 or do we keep it as 0/e?

5. Jul 6, 2010

### Staff: Mentor

I agree with Bohrok and disagree with njama. Let's take it a step at a time. Your first differentiation of the numerator was correct, but the second one wasn't, and your latest try doesn't look right either.

What do you get for
$$\frac{d}{dx} 2xe^{x^2}~cos(e^{x^2} - 1)$$ ?

Also, you should NOT leave an answer as 0/e.

6. Jul 6, 2010

### RockPaper

Thank you for clearing that up, so the limit as x->0 is 0

I get:
2ex2+4x2ex2cos(ex2-1)-sin(ex2-1)*2xex2*2xex2

7. Jul 6, 2010

### Staff: Mentor

Close, but you are missing a pair of necessary parentheses and one exponent is incorrect, according to my work.
Edit: the 3 exponent below should be 2.
(2ex2+4x3ex2)cos(ex2-1)-sin(ex2-1)*2xex2*2xex2

BTW
$$2xe^{x^2}*2xe^{x^2} = 4x^2e^{2x^2}$$

Last edited: Jul 6, 2010
8. Jul 6, 2010

### RockPaper

I'm glad I finally got it because it took me a few tries to get it :/ which is typical.
I don't understand how it's 4x3.
And thank you for telling me that it's 4x^2e^(2x^2) because I didn't know how to simplify it :shy:

9. Jul 6, 2010

### Staff: Mentor

You're right - that exponent shouldn't be 3. It took me several times checking to find what I did wrong.

$$d/dx(2xe^{x^2}~cos(e^{x^2} - 1)) = 2xe^{x^2}(-sin(e^{x^2} - 1) \cdot 2xe^{x^2}) + d/dx(2xe^{x^2}) \cdot cos(e^{x^2} - 1)$$
$$=-4x^2e^{2x^2}sin(e^{x^2} - 1) + (2e^{x^2} + 2x \cdot 2xe^{x^2})cos(e^{x^2} - 1)$$
$$= (2e^{x^2} + 4x^2e^{x^2})cos(e^{x^2} - 1) - 4x^2e^{2x^2} sin(e^{x^2} - 1)$$

10. Jul 6, 2010

### RockPaper

Great! Thank you for the help... I actually added the parenthesis on my paper but somehow didn't do that when I wrote it on the computer. :| It took me a whole bunch of tries to get this right (even when I had an idea on what to do), I need to work on my math skills which I'm hoping to get better over the summer. Hopefully I'll finally gain full understanding on math once I practice and learn the rules.