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

[riri]

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 $$\frac{cos5x-cos6x}{x^2}$$

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!

 

[Greg]

$$\lim_{x\to0}\dfrac{\cos(5x)-\cos(6x)}{x^2}$$

Differentiate numerator and denominator once:

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

Differentiate again:

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

 

[Prove It]

$$\lim_{x\to0}\dfrac{\cos(5x)-\cos(6x)}{x^2}$$

Differentiate numerator and denominator once:

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

Differentiate again:

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

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