Understanding L'Hospital's Rule for Calculating Limits

[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.)

 

[ShayanJ]

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

 

[micromass]

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!).

 

[SammyS]

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!).

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.

 

[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.

 

[ShayanJ]

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

 

[micromass]

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.

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.

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.

Can you not simplify the denominator?

 

[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!