L'Hopital's Rule and Infinite Limits

[bombadil]

Say you have a limit in indeterminate form (0/0 or infinity/infinity) and you apply L'Hopital's rule to it and the result is an infinite limit. Is that a valid answer? Can L'Hopital's rule be applied in this way?

[micromass]

Yes, that is valid.

Can you give us the limit to make sure we mean the same thing here??

[bombadil]

Here's the limit I'm thinking of:



\lim_{\substack{R\rightarrow 1}} \frac{RP'}{P},



where primes are derivatives w.r.t. R. Also,



P= c R J_1(\alpha R) - \frac{R^2 F}{\alpha^2},



where J_1 is a Bessel function of the first kind. Two of the three constants (c,alpha,F) are chosen such that P(1)=0 and P'(1)=0 and the third is chosen for convenience. Thus the limit is in the form 0/0, so L'Hopital's rule leads to the following:



\lim_{\substack{R\rightarrow 1}} \frac{RP'}{P}=\left[1+R\frac{P''}{P'}\right]_{R=1}\rightarrow \infty

[micromass]

Ah yes. What you did is indeed a valid use of L'Hopitals rule.