# Limit change of variable legality?

1. ### Pengwuino

6,949
1. The problem statement, all variables and given/known data

$$\mathop {\lim }\limits_{x - > 1} \ln (1 - x)(P_{l - 1} (x) - xP_l (x)) - \mathop {\lim }\limits_{x - > - 1} \ln (1 + x)(P_{l - 1} (x) - xP_l (x))$$

where $$P_{l}$$ are the Legendre Polynomials.

2. Relevant equations

Legendre polynomial recursion relations I suppose

3. The attempt at a solution

So in trying to determine this limit. L'Hospitals rule doesn't come up with anything useful seemingly. However, would it be mathematically legal to switch the limit of the right side to x-> 1 and switch out every x for -x? I can't seem to figure out this limit. If this were legal, I think I could determine this limit for l = 0 using this idea but for an arbitrary l I don't think this would work. Anyone got any tips? :)

2. ### Gib Z

3,347
Basically you want to make the substitution u= -x. I think as long as the substitution is a continuous (perhaps monotonic as well?) function then the limit is still valid. In other words, yea it'll work.

3. ### Pengwuino

6,949
Hmmm ok. I suppose the only problem now is making the substitution into the Legendre Polynomials which I think I can do to turn it all into 1 limit that subtracts away the logarithms that blow up on me.