Trying to fix a simplification issue for a general range formula

[FallenLeibniz]

So I have derived a formula for the range of projectile in a vacuum fired from an arbitrary height h from above the ground as the following...

R=(vcosθ/g)[vsinθ+sqrt((vsinθ)^2+2gh)]Now it seems all well and good, but when I try to play with it by multiplying the RHS by the congugate term over the conjugate term (i.e. multiplying by one), I keep coming up with the 2gh factor on top which if h=0 leads to a discrepancy between the two. Can anyone help reconcile this? Note that all factors in the equation are positive values. I can provide a derivation of this equation later if necessary, but am late to be somewhere and so I must do it later.

 

[haruspex]

It's not there's a discrepancy. If you plug in h = 0 you will 0/0, which is indeterminate. So the right thing to do is take the limit as h tends to 0 from above, and that will give you sin(2θ)/g, as before.

 

[FallenLeibniz]

Ok, I see the discontinuity now. I feel horrifically blind now.

 

[FallenLeibniz]

Thank you.