L'Hospital Q: Is it Valid for "Regular" Limits?

  • Thread starter EvLer
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In summary, the use of L'Hopital's rule is only valid for limits of indeterminate form, such as 0/0 or inf/inf. It cannot be applied to regular limits, even if it seems to give the same answer. This rule states that the ratio of limits when they are 0 or inf is equal to the ratio of their derivatives. However, this cannot be applied to a regular limit, as shown in the example of the limit of (-sin(x))/1 as x approaches 0.
  • #1
EvLer
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I was just wondering... it is used when the limit is of indetermined form: 0/0 or inf/inf, but is it valid to apply it to a "regular" limit? technically it seems to give the same answer...
 
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  • #2
If I recall, his rule was to the effect that the ratio of the limits, when the limits are 0 or [itex]\infty[/itex], is the ratio of the derivatives. That said, you certainly cannot apply it to a regular limit.

As an example, consider the limit:

Lim x->0 (cos(x)+3) /(x+4)

It is clear that the above limit is 4/5.

On the other hand,

Lim x-> 0 (-sin(x))/(1) = 0.

Carl
 
  • #3
Oh, ok thanks...
 

1. Is L'Hospital's rule only valid for "regular" limits?

Yes, L'Hospital's rule is only valid for "regular" limits, which are limits where both the numerator and denominator approach a constant value as the independent variable approaches a specific value.

2. What is the purpose of L'Hospital's rule?

The purpose of L'Hospital's rule is to simplify the evaluation of indeterminate forms (such as 0/0 or ∞/∞) in calculus by finding the limit of the ratio of the derivatives of the numerator and denominator.

3. 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 as long as the limit is in the form of 0/0 or ∞/∞ and both the numerator and denominator are differentiable.

4. Are there any limitations to using L'Hospital's rule?

Yes, L'Hospital's rule cannot be used to evaluate limits that do not approach a constant value, such as limits that approach infinity or negative infinity. It also cannot be used for limits involving complex numbers or limits involving functions with discontinuities.

5. Can L'Hospital's rule be applied multiple times?

Yes, L'Hospital's rule can be applied multiple times as long as the resulting limit is still in indeterminate form. However, it is important to note that blindly applying L'Hospital's rule multiple times may lead to incorrect results, so it should be used with caution.

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