# Implementation of Series function

I'm trying to evaluate the partial derivatives of some function F[x,y] at some point, i.e.

Limit[Limit[D[F[x,y],{x,m},{y,n}],x->x0],y->y0]


However, for higher derivatives I find that it fails to evaluate the limit. On the other hand, if I use the Series function

Series[F[x,y],{x,x0,m},{y,y0,n}]


It produces all the coefficients with no problem. My question is what is the difference between the two methods? what is Series assuming that I'm missing when I do it manually with the derivatives?

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What's a good example for F? I see no difference for, say, Sin[x] Exp[y]. – Michael E2 Aug 28 '14 at 11:29
@MichaelE2 Example Limit[D[BesselJ[c, x]/x, {x, 3}], x -> 0] vs Series[BesselJ[c, x]/x, {x, 0, 3}] The former times out while the latter gives the right result by looking at the coefficient of x^3 – Prastt Aug 28 '14 at 18:31
From the result of Trace, it seems Series[BesselJ[c, x]/x, {x, 0, 3}] is calculated by calculating Series[1/x, {x, 0, 3}] * Series[BesselJ[c, x], {x, 0, 3}]. – Silvia Sep 3 '14 at 0:14