# Limits applying L'Hospital

1. Aug 31, 2014

### Akitirija

$$\lim_{x\to 0} \frac{x}{\sqrt{4+x}-\sqrt{4-x}}=2$$

I plotted this equation into WolframAlpha, and it applies L'Hospital and yields, for the nominator:

$$\frac{dx}{dx}=2$$

I do not understand this. I have not learned how to use L'Hospital, but I know basic derivation rules, and I do not understand how this is correct. Any hints?

(The problem is 1.2-25 from Calculus: A Complete Course, 7th Edition by Robert A. Adams.)

2. Aug 31, 2014

### ShayanJ

You should differentiate nominator and denominator separately, as if they were two unrelated functions. Then plug in x=0.

3. Aug 31, 2014

### micromass

Staff Emeritus
Multiply both numerator and denominator by

$$\sqrt{4 + x} + \sqrt{4 - x}$$

As an aside, I'm not a fan of L'Hospital's rule. I feel that it is often the lazy way out. Proving a limit wihtout L'Hospital is often much more fun (and hard!).

4. Aug 31, 2014

### SammyS

Staff Emeritus
micromass makes an excellent point here. There is also the mistake of using L'Hôpital's rule erroneously in cases to which it does not apply !

However, if you're learning to use L'Hôpital's rule, then use it. Do the problem in the manner suggested by micromass as a check on your result.

5. Aug 31, 2014

### Akitirija

Thank you, Shyan and micromass!

micromass, I try to avoid L'Hospital (because I do not think one should use a method that one cannot prove), but I thought that one sometimes had to use it, so I am very happy about your answer.

I have to admit, though, that I am still struggling with the same calculation. Probably my algebra is too rusty. When I do what you suggested, this is what I get:

$$\lim_{x\to 0} \frac{x(\sqrt{4+x}+\sqrt{4-x})}{(4+x)-(4-x)}$$

And I have no idea what to do from here.

6. Aug 31, 2014

### ShayanJ

The proof of L'Hôpital's rule is pretty easy.
Take a look at here.

7. Aug 31, 2014

### micromass

Staff Emeritus
That is a very good attitude. Especially when starting out with mathematics, using something you haven't proven is a big no no. Of course, one must be able to relax this rule too. It is not possible to prove everything in an intro class and you'll need to be able to apply certain techniques whose proof you'll only encounter later. The technique of de L'Hospital is one of those techniques (although its proof is not that difficult). But still, proving things without de L'Hospital can be a wonderful and fruitful exercise.

Can you not simplify the denominator?

8. Aug 31, 2014

### Akitirija

micromass, thank you so much!

That's embarrassing, I think I did too many calculations in one day and just suddenly turned blind!

I appreciate your help a lot! Thank you again!

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