L'Hopital's Rule for Solving Limits

In summary, the conversation discusses using L'Hopital's rule to find the limit of (tan(x)-x)/(sin2x-2x) as x approaches 0. The person asking for help has taken the derivative multiple times but is still unable to reach the correct answer. The expert suggests using the product rule on the numerator and provides the correct solution of -1/4.
  • #1
moloko
5
0

Homework Statement


lim as x->0 (tan(x)-x)/(sin2x-2x)


Homework Equations


L'Hopitals rule states that if the limit reaches 0/0, you can take the derivative of the top and the bottom until you get the real limit.


The Attempt at a Solution



(sec^2(x)-1)/(2cos2x-2) still 0/0
2sec^2(x)tan(x)/(-4sin(2x)) still 0/0

I have pain stakingly taken the derivative twice more and it simply does not seem to reach any end. All help is very much appreciated!
 
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  • #2
You are going to get a nonzero denominator at the next derivative after you've shown. It's a cosine.
 
  • #3
You are correct and this does give me a non 0/0 answer but the answer is wrong according to the book. It should be -1/4.

My next derivative is:

(2*(2sec^2(x)tan(x))*tan(x))/(-8cos(2x))

In simpler terms

(4sec^2(x)tan^2(x))/(-8cos(2x))

The numerator becomes 0 with the limit and the demoninator becomes -8, making the limit 0.

Thanks
 
  • #4
Why aren't you using the product rule on the numerator? What ARE you doing? The derivative of the tan(x) term will be nonzero.
 
  • #5
I apologize, that was a careless mistake.

I'll post my solution if it's of any help to anyone...

the numerator becomes 2(sec^2(x)tan^2(x) + sec^2(x)),
which of course goes to 2 when pushed to 0
leaving 2/-8, or -1/4

Thank you so much!
 

1. What is L'Hopital's Rule?

L'Hopital's Rule is a mathematical theorem that helps to evaluate the limit of a function in cases where the limit is of the form "0/0" or "∞/∞". It provides a method for finding the limit by taking the derivative of the top and bottom of the fraction separately and then evaluating the limit again.

2. When should I use L'Hopital's Rule?

L'Hopital's Rule should only be used when the limit is of the form "0/0" or "∞/∞". It cannot be applied to other types of limits.

3. What are the steps for using L'Hopital's Rule?

The steps for using L'Hopital's Rule are as follows:
1. Simplify the expression and rewrite it in the form of "0/0" or "∞/∞".
2. Take the derivative of the numerator and denominator separately.
3. Evaluate the limit again using the new derivatives.
4. If necessary, repeat the process until the limit can be evaluated.

4. What are the limitations of L'Hopital's Rule?

L'Hopital's Rule can only be applied to limits of the form "0/0" or "∞/∞". It cannot be used to evaluate other types of limits. Additionally, it may not work if the limit is indeterminate, such as "0*∞" or "∞-∞". Lastly, L'Hopital's Rule only applies to single variable limits and cannot be used for multivariable limits.

5. Are there any alternatives to using L'Hopital's Rule?

Yes, there are alternative methods for evaluating limits of the form "0/0" or "∞/∞". One method is to use algebraic manipulation to simplify the expression and then evaluate the limit. Another method is to use the Squeeze Theorem, which states that if a function is between two other functions that have the same limit at a point, then the function in the middle must also have the same limit at that point.

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