Derivative using l'hopital's rule

In summary, l'Hopital's rule is a mathematical method used to evaluate limits involving indeterminate forms. It should only be used when evaluating limits that result in indeterminate forms, and it involves taking the derivative of both the numerator and denominator of the quotient. However, there are limitations to its use, as it cannot be used for limits that do not result in indeterminate forms, improper limits, or functions that are not differentiable.
  • #1
losin
12
0
let f(x)

exp(-1/x) for x>0, 0 for x<=0

i want to get f'(x) by using l'hopital's rule, but somehow

i'm applying l'hopital's rule again and again and no clear value is coming out.

i know f'(0) is 0, but i cannot prove it
 
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  • #2
So, using the definition of the derivative, you're looking for the limit as x-->0 of exp(-1/x)/x.

If you use L'Hopital on this, you get that it is equal to the limit as x-->0 of -exp(-1/x)/x.

So...?
 

FAQ: Derivative using l'hopital's rule

What is l'Hopital's rule?

L'Hopital's rule is a mathematical method used to evaluate limits involving indeterminate forms, where the limit of a quotient is not immediately obvious. It states that if the limit of the quotient of two functions exists, then the limit of the quotient of their derivatives also exists and is equal to the limit of the original quotient.

When should l'Hopital's rule be used?

L'Hopital's rule should only be used when evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It cannot be used for limits that do not result in indeterminate forms.

How do you apply l'Hopital's rule?

To apply l'Hopital's rule, first determine if the limit is an indeterminate form. If it is, take the derivative of both the numerator and denominator of the quotient. Then, evaluate the limit of the new quotient. If the new limit is still an indeterminate form, you can continue taking derivatives until the limit is no longer indeterminate.

What are the limitations of l'Hopital's rule?

L'Hopital's rule cannot be used to evaluate limits that do not result in indeterminate forms. It also cannot be used to evaluate improper limits, such as those involving infinity or negative infinity. Additionally, it should only be used when taking the derivative of the numerator and denominator will not result in an endless loop.

Can l'Hopital's rule be used for all types of functions?

No, l'Hopital's rule can only be used for functions that are differentiable. This means that the functions must have a well-defined derivative at the point where the limit is being evaluated. It cannot be used for functions that are not continuous or not differentiable at that point.

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