# L'Hôpital's rule for infinity minus infinity

1. Oct 16, 2011

### swampwiz

It seems that this particular case of L'Hôpital's rule is really of the form of a pair of quotient functions (i.e., each function having a dividend & divisor) such that the divisor for each is zero, and the dividend for each is not zero, such that the quotient for each is infinity. With that in mind, the idea is to create a new quotient function which is equivalent to the original subtraction of the original quotient functions. Because the divisor for each original quotient is zero, the new dividend and divisor must be zero. From here, it is just a standard application of the rule to zero divided by zero.

Is this accurate? Are there any other situations that would suffice for infinity minus infinity? Thanks

f = p / q
g = r / s

q = s = 0 → f = g = ∞

u = f s - g q = 0
v = q s = 0

f - g = u / v = u' / v'