Example of L'Hopital's Rule (Hard)

Calculus: We compute the limit of (1-cos(x))^3/2 /sin(x) as x goes to zero. A straightforward approach with multiple applications of L'Hopital's rule fails. We consider options, based on trig identities and square roots.

[krlsjke]

I appreciate your highly coherent teaching style and well organized written material!

[MathDoctorBob]

Your welcome! For the 0/0 case, the proof is a straight-forward application of the Mean Value Theorem. To believe it should work: assume the functions have convergent Taylor series expansions at x=c. By assumption, the constant terms a0, b0 = 0 and we can cancel an (x-c) in the quotient. When we evaluate at x=c, the quotient now becomes a1/b1, which is f'(c)/g'(c).

[MathDoctorBob]

Thanks! This one turned out better than planned (with an obligatory typo). Of course the best ideas come a few days after posting.

[MathDoctorBob]

I have issues with English (native), let alone French. I'm still trying to figure out how "Zero comes in from the right" works. :)

[sunshinedaydream]

Thank you for the video. I'm still pre-undergrad but I'm looking forward to learning more about L'Hopital's rule, seems very useful. The thing which always bothers me is not knowing a formal proof of it when using it!

[Alexis-ttbl]

can you please do all the algebra next time this is very confusing

[thebgEntertainment]

replacing 1-cos(x) and x-sin(x) with their taylor polynomials makes this really easy

[dasdos]

The H is silent. But great video and good problem!

[puffpuffpass]

u should train how to work out instead