Understanding L'Hospital's Rule for Calculating Limits

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In summary, the conversation discusses the use of L'Hospital's rule to solve a limit problem involving the equation 1.2-25 from Calculus: A Complete Course, 7th Edition by Robert A. Adams. However, one person expresses their discomfort with using a method they cannot prove. The other person advises to prove the method, but also acknowledges that sometimes it may be necessary to use it without proof. Eventually, the person struggling with the calculation realizes their mistake and thanks the other person for their help.
  • #1
Akitirija
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$$\lim_{x\to 0} \frac{x}{\sqrt{4+x}-\sqrt{4-x}}=2$$


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.)
 
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  • #2
You should differentiate nominator and denominator separately, as if they were two unrelated functions. Then plug in x=0.
 
  • #3
Multiply both numerator and denominator by

[tex]\sqrt{4 + x} + \sqrt{4 - x}[/tex]

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!).
 
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  • #4
micromass said:
Multiply both numerator and denominator by

[tex]\sqrt{4 + x} + \sqrt{4 - x}[/tex]

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.
 
  • #5
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
The proof of L'Hôpital's rule is pretty easy.
Take a look at here.
 
  • #7
Akitirija said:
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?
 
  • #8
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!
 

Related to Understanding L'Hospital's Rule for Calculating Limits

1. What is L'Hospital's Rule?

L'Hospital's Rule is a mathematical technique used to evaluate limits of indeterminate forms, where both the numerator and denominator approach zero or infinity. It involves taking the derivative of the numerator and denominator separately and then evaluating the limit again.

2. When can L'Hospital's Rule be used?

L'Hospital's Rule can only be used when the limit is in an indeterminate form, such as 0/0 or ∞/∞. It cannot be used when the limit is already in a determinate form, such as 5/2 or ∞/3.

3. How do you apply L'Hospital's Rule?

To apply L'Hospital's Rule, take the derivative of the numerator and denominator separately, then simplify the resulting expression. If the resulting expression is still in an indeterminate form, repeat the process until the limit can be evaluated.

4. Are there any restrictions when using L'Hospital's Rule?

Yes, there are some restrictions when using L'Hospital's Rule. The limit must have the form 0/0 or ∞/∞ and both the numerator and denominator must be differentiable functions. Additionally, the limit must approach a single value, not different values from the left and right sides.

5. Can L'Hospital's Rule be used for limits involving trigonometric functions?

Yes, L'Hospital's Rule can be used for limits involving trigonometric functions. However, the limit must first be rewritten in a form where the numerator and denominator are both a quotient of two functions. Then, L'Hospital's Rule can be applied to evaluate the limit.

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