I've been reading about partial fraction expansions using L'Hopital's Theorem and have found this document.
I was wondering if it were possible to use the method mentioned in the link to find all the coefficients of a function with a perfect square as the denominator.
For example, let us suppose I had: $ \frac{x+1}{(x+7)^{2}} = \frac{A}{(x+7)^{2}} + \frac{B}{x+7}$
I can find $A$, the problem is, finding $B$.
This is my attempt: $ B = \lim_{x\to-7} \frac{(x+1)(x+7)}{(x+7)^{2}} + \lim_{x\to-7} \frac{A(x+7)}{(x+7)^2} $ $ B = (-7+1)\lim_{x\to-7} \frac{(x+7)}{(x+7)^{2}} + A\lim_{x\to-7} \frac{(x+7)}{(x+7)^2} $
So the problem is, the degree of the denominator is greater than that of the numerator, and if I use L'Hopital's Theorem, I can never make it so that I can sub in $-7$ without getting an undefined value.
I looked up the answer online and $ A = -6 $ and $ B = 1 $.
I should also mention, this is not at all homework, I'm just reading about partial fractions and I wondered about this.
Thanks.