Solve L'Hospital's Rule Homework with 0/0

  • Thread starter tornzaer
  • Start date
In summary, applying L'Hopital's rule twice gives the correct answer for the first derivative, but results in an incorrect answer for the second derivative.
  • #1
tornzaer
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0

Homework Statement



lim -> 0 (sin(x)-x) / x^3

Homework Equations



L'Hospital's Rule

The Attempt at a Solution



I could just use L'Hospital's rule since it's 0/0. However, the answer is wrong when I do it that way. What am I missing? It states on the question that it must be rewritten before the ruel can be applied and that it has to be applied more than once.

Please help.
 
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  • #2
will have a closer look, but one way to gain better understading about the problem would be to expand the sin in a taylor series about 0...
 
  • #3
show your L'Hop working, i think it works fine, but has to be applied twice
 
  • #4
I don't see why you'd need to rewrite anything. You just have to argue that

lim x->0 f/g = lim x->0 f'/g' = lim x->0 f''/g'' = ... is true for the nth derivate as long as the limit of n-1th derivate is either zero or infinite for both f and g.
 
  • #5
The first time applied gives me: (cosx - 1) / 3x^2

The second: (-sinx) / 6x

I definitely know this is wrong...
 
  • #6
... and what would the third give you?
 
  • #7
Third would be -cosx/6
 
  • #8
lanedance said:
will have a closer look, but one way to gain better understading about the problem would be to expand the sin in a taylor series about 0...

Pretty much what applying L'Hopital's rule does really.
 
  • #9
tornzaer said:
Third would be -cosx/6
And the limit of that as x goes to 0 is ?
 
  • #10
Try expanding sin(x) using the Maclaurin series.
You will get the answer directly.
 

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  • #11
HallsofIvy said:
And the limit of that as x goes to 0 is ?

That's not the answer according to calculators.
 
  • #12
tornzaer said:
That's not the answer according to calculators.

According to Mathematica it is.
 
  • #13
tornzaer said:
That's not the answer according to calculators.

statements like that, with very little information, are hard to help with...

show your method & results and explain why you think there is a disconnect, and we can comment/help out...
 

What is L'Hospital's Rule?

L'Hospital's Rule is a mathematical theorem that helps evaluate limits involving indeterminate forms, such as 0/0 or infinity/infinity.

When should I use L'Hospital's Rule?

L'Hospital's Rule should only be used when the limit of a function is in an indeterminate form, such as 0/0 or infinity/infinity.

How do I use L'Hospital's Rule to solve a limit with 0/0?

To solve a limit with 0/0 using L'Hospital's Rule, take the derivative of the numerator and denominator separately, then evaluate the limit again. If the limit is still in an indeterminate form, repeat the process until the limit can be evaluated.

What are the limitations of L'Hospital's Rule?

L'Hospital's Rule can only be used for certain types of limits and may not always give the correct answer. It also cannot be used for limits that tend to infinity or negative infinity.

Is there an alternative to using L'Hospital's Rule for limits with 0/0?

Yes, there are other methods for evaluating limits with 0/0 such as factoring, algebraic manipulation, or using known limits. It is important to try these methods first before resorting to L'Hospital's Rule.

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