L'Hopital's Rule: Evaluating Limits

In summary, the conversation discusses using l'Hopital's rule to evaluate a limit involving e^(-1/x) and x, and clarifies that l'Hopital's rule can also be used for limits that do not fall into the category of indeterminate forms. The limit in question approaches a very small number as x approaches 0 from the positive side.
  • #1
SherlockOhms
310
0

Homework Statement



Use l'Hopital's rule to evaluate the following limit:
lim x→0 e^(-1/x) / x for x> 0.

Homework Equations



differentiate the top and bottom until a limit can be found. Possibly rewrite as a product.

The Attempt at a Solution


I was under the impression that l'Hopital's rule could only be used for evaluating limits of indeterminate form i.e. 0/0 or ∞/∞. The above quotient doesn't fall into this category, does it? If someone could clear this up for me it'd be great.
 
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  • #2
DAPOS said:

Homework Statement



Use l'Hopital's rule to evaluate the following limit:
lim x→0 e^(-1/x) / x for x> 0.

Homework Equations



differentiate the top and bottom until a limit can be found. Possibly rewrite as a product.

The Attempt at a Solution


I was under the impression that l'Hopital's rule could only be used for evaluating limits of indeterminate form i.e. 0/0 or ∞/∞. The above quotient doesn't fall into this category, does it? If someone could clear this up for me it'd be great.
What is ##\lim_{x \to 0^+} e^{-1/x}##?
 
  • #3
jbunniii said:
What is ##\lim_{x \to 0^+} e^{-1/x}##?

It approaches some very small number. Ok, I see now. Thanks!
 

1. What is L'Hopital's Rule?

L'Hopital's Rule is a mathematical theorem that helps evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of a function f(x) and g(x) as x approaches a is both 0 or both ∞, then the limit of their quotient (f(x)/g(x)) can be found by taking the derivative of both f(x) and g(x) and then evaluating the limit again.

2. When is L'Hopital's Rule applicable?

L'Hopital's Rule can only be applied when both the numerator and denominator of a limit are in an indeterminate form, such as 0/0 or ∞/∞. It is also applicable when the limit involves exponential or logarithmic functions.

3. How do you use L'Hopital's Rule to evaluate a limit?

To use L'Hopital's Rule, you first need to determine if the limit is in an indeterminate form. If it is, then take the derivative of the numerator and denominator separately, and then evaluate the limit again. If the new limit is still in an indeterminate form, then repeat the process until the limit can be determined.

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

Yes, there are a few restrictions when using L'Hopital's Rule. First, the limit must be in an indeterminate form. Additionally, the functions in the limit must be differentiable at the point a. Finally, the limit must exist in order for L'Hopital's Rule to be applicable.

5. Can L'Hopital's Rule be used to evaluate limits at infinity?

Yes, L'Hopital's Rule can also be used to evaluate limits at infinity. In this case, the limit would involve a function as x approaches ∞ or -∞. The same process of taking the derivative of the numerator and denominator and evaluating the limit again applies in this case as well.

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