Solving Limits: l'Hopital's Rule & Degree Rule

The degree rule does not apply here.In summary, the limit of \frac{cos5x-cos6x}{x^2} as x approaches 0 is \frac{11}{2}. The l'hopital's rule was used to differentiate the numerator and denominator, and the degree rule does not apply in this case.
  • #1
riri
28
0
Hi, I'm having some trouble with finding the limit for this question:

I can use the l'hopital's rule which I tried.. I tried pi, 2pi, 0, inf, none seem to work so if I could have some help that would be appreciated!

limx→0 \(\displaystyle \frac{cos5x-cos6x}{x^2}\)

Would the degree rule apply here? It wouldn't just be the 5x/x^2 and 6x/x^2 right? because that would give me 0...

Thanks!
 
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  • #2
\(\displaystyle \lim_{x\to0}\dfrac{\cos(5x)-\cos(6x)}{x^2}\)

Differentiate numerator and denominator once:

\(\displaystyle \lim_{x\to0}\dfrac{-5\sin(5x)+6\sin(6x)}{2x}\)

Differentiate again:

\(\displaystyle \lim_{x\to0}\dfrac{-25\cos(5x)+36\cos(6x)}{2}=\dfrac{11}{2}\)
 
  • #3
greg1313 said:
\(\displaystyle \lim_{x\to0}\dfrac{\cos(5x)-\cos(6x)}{x^2}\)

Differentiate numerator and denominator once:

\(\displaystyle \lim_{x\to0}\dfrac{-5\sin(5x)+6\sin(6x)}{2x}\)

Differentiate again:

\(\displaystyle \lim_{x\to0}\dfrac{-25\cos(5x)+36\cos(6x)}{2}=\dfrac{11}{2}\)

Of course, this method only works because you have 0/0 indeterminate forms.
 

1. What is l'Hopital's Rule and when should it be used?

L'Hopital's Rule is a mathematical tool used to solve limits involving indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of the quotient of two functions is indeterminate, then the limit is equal to the quotient of the derivatives of the two functions. It should be used when direct substitution or other algebraic techniques fail to solve the limit.

2. What is the Degree Rule and how is it used to solve limits?

The Degree Rule is a simplification technique used to solve limits involving polynomials. It states that if the limit expression is in the form of a fraction where the degree of the numerator is greater than the degree of the denominator, then the limit is equal to either positive or negative infinity, depending on the signs of the leading coefficients. This rule is used to quickly determine the end behavior of a polynomial function and the corresponding limit.

3. Can l'Hopital's Rule be used for limits at infinity?

Yes, l'Hopital's Rule can be used for limits at infinity. In this case, the limit expression is written in the form of a quotient where both the numerator and denominator go to infinity. The rule can then be applied by taking the derivatives of the numerator and denominator and finding the limit of the resulting quotient. This is useful for evaluating the horizontal asymptotes of a function.

4. Are there any limitations to using l'Hopital's Rule?

Yes, there are certain limitations to using l'Hopital's Rule. It can only be applied when the limit expression is in an indeterminate form, and the derivatives of the functions involved must exist. Additionally, it should only be used as a last resort when other techniques fail to solve the limit. It is also important to check for any discontinuities or other special cases before applying the rule.

5. Can the Degree Rule be used for limits of trigonometric functions?

No, the Degree Rule cannot be used for limits of trigonometric functions. This rule is specific to polynomials and cannot be applied to trigonometric expressions. Instead, other techniques such as trigonometric identities and algebraic manipulation should be used to solve these types of limits.

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