Its to handle expressions such as this, in respect to variable ao:

solu7 = -((ao^3 (r2 r3 rb rc + 
r1 (r4 ra rc + r3 rb rc + 
r3 ra (rb + rc))) rgsink vref + (r1 + r2) (r3 + 
r4) (ra rc rgsink + rb rc rgsink + ra rb (rc + rgsink)) vth + 
ao (r2 (r4 ra rb rgsink vref + r3 rb rc rgsink vth + 
r3 ra (rb rc vth + rc rgsink vth + 
rb rgsink (vref + vth))) + 
r1 (2 r3 rb rc rgsink vth + r4 rb rc rgsink vth + 
r4 ra (rb rc vth + rc rgsink vth + 
rb rgsink (vref + vth)) + 
r3 ra (2 rb rc vth + 2 rc rgsink vth + 
rb rgsink (vref + 2 vth)))) + 
ao^2 (r2 (r3 rb rc + r4 (ra + rb) rc + 
r3 ra (rb + rc)) rgsink vref + 
r1 (r4 (rb rc + ra (rb + rc)) rgsink vref + 
r3 (ra rc rgsink (vref + vth) + rb rc rgsink (vref + vth) +
ra rb (rc vth + 
rgsink (2 vref + vth))))))/(r2 r4 (-((-1 + 
ao^2) ra rc rgsink) - (-1 + ao^2) rb rc rgsink + 
ra rb (rc + rgsink - ao rgsink)) + 
r2 r3 ((1 + ao - ao^2) ra rc rgsink - (-1 + ao) (1 + 
ao)^2 rb rc rgsink + (1 + ao) ra rb (rc + rgsink - 
ao rgsink)) + (1 + 
ao) r1 r3 ((1 + ao - ao^2) ra rc rgsink + (1 + ao - 
ao^2) rb rc rgsink + (1 + ao) ra rb (rc + rgsink - 
ao rgsink)) + 
r1 r4 ((1 + ao - ao^2) rb rc rgsink + (1 + 
ao) ra (-((-1 + ao^2) rc rgsink) + 
rb (rc + rgsink - ao rgsink)))))

The funny thing is with the function definition I'm working with:

lhospital = 
Function[{f, x}, 
Switch[Head[Unevaluated[f]], Times, 
Divide @@ D[NumeratorDenominator[Unevaluated[f]], x]], HoldAll]

(I used Switch, so I can keep adding more cases as more heads needs to be handled or extended to)

This works:

lhospital[-((ao^3 (r2 r3 rb rc + 
r1 (r4 ra rc + r3 rb rc + 
r3 ra (rb + rc))) rgsink vref + (r1 + r2) (r3 + 
r4) (ra rc rgsink + rb rc rgsink + ra rb (rc + rgsink)) vth + 
ao (r2 (r4 ra rb rgsink vref + r3 rb rc rgsink vth + 
r3 ra (rb rc vth + rc rgsink vth + 
rb rgsink (vref + vth))) + 
r1 (2 r3 rb rc rgsink vth + r4 rb rc rgsink vth + 
r4 ra (rb rc vth + rc rgsink vth + 
rb rgsink (vref + vth)) + 
r3 ra (2 rb rc vth + 2 rc rgsink vth + 
rb rgsink (vref + 2 vth)))) + 
ao^2 (r2 (r3 rb rc + r4 (ra + rb) rc + 
r3 ra (rb + rc)) rgsink vref + 
r1 (r4 (rb rc + ra (rb + rc)) rgsink vref + 
r3 (ra rc rgsink (vref + vth) + rb rc rgsink (vref + vth) +
ra rb (rc vth + 
rgsink (2 vref + vth))))))/(r2 r4 (-((-1 + 
ao^2) ra rc rgsink) - (-1 + ao^2) rb rc rgsink + 
ra rb (rc + rgsink - ao rgsink)) + 
r2 r3 ((1 + ao - ao^2) ra rc rgsink - (-1 + ao) (1 + 
ao)^2 rb rc rgsink + (1 + ao) ra rb (rc + rgsink - 
ao rgsink)) + (1 + 
ao) r1 r3 ((1 + ao - ao^2) ra rc rgsink + (1 + ao - 
ao^2) rb rc rgsink + (1 + ao) ra rb (rc + rgsink - 
ao rgsink)) + 
r1 r4 ((1 + ao - ao^2) rb rc rgsink + (1 + 
ao) ra (-((-1 + ao^2) rc rgsink) + 
rb (rc + rgsink - ao rgsink))))), ao]

But this doesn't:

lhospital[solu7, ao]

Power::infy: Infinite expression 1/0 encountered.
ComplexInfinity

When numerator & denominator both still have ao as polynomials (therefore going to infinity).