Limits: Finding the Value of (x - Sinx) / x^3

In summary, the conversation discusses the method of using L'Hopital's Rule to find the limit of (x-Sinx)/x^3 as x approaches 0, which is equal to 1/6. Other possible approaches, such as using the Maclaurin series, are mentioned but may not be applicable if the person is not familiar with L'Hopital's Rule or series.
  • #1
rambo5330
84
0

Homework Statement


lim (x->0) of (x - Sinx) / x^3


Homework Equations





The Attempt at a Solution


I would like to know the proper route for analyzing that question... the text is saying i should end up with 1/6 ...however I'm not sure how they arrived at that answer ...
 
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  • #2
Are you familiar with l'hopitale's rule?

if lim (x->0) a/b = 0/0 or infinity/infitity then lim (x->0) a/b = lim (x->0) a'/b'
 
  • #3
not familiar with that yet ...
 
  • #4
Another approach is to replace sin x by a few terms of its Maclaurin series, but if you don't know L'Hopital's Rule, you probably don't know anything about series yet, either. Short of those two techniques, I can't think of anything else that would work for you.
 

FAQ: Limits: Finding the Value of (x - Sinx) / x^3

1. What is the limit of (x - Sinx) / x^3 as x approaches 0?

As x approaches 0, the limit of (x - Sinx) / x^3 is 1/6.

2. How do you find the limit of a function?

To find the limit of a function, you can evaluate the function as x approaches the given value. If the function is undefined at that value, you can use algebraic manipulations or L'Hopital's rule to simplify the expression and find the limit.

3. Can the limit of a function exist even if the function is undefined at that point?

Yes, a function can have a finite limit even if it is undefined at that point. This is known as a removable discontinuity.

4. What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit only considers the behavior of the function as x approaches the given value from one direction (either from the left or the right). A two-sided limit considers the behavior of the function from both directions.

5. How can you use a graph to determine the limit of a function?

You can use a graph to estimate the limit of a function by observing the behavior of the function as x approaches the given value. If the function approaches a specific value or has a horizontal asymptote, that value would be the limit.

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