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.