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Re: Wrong limit?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104691] Re: Wrong limit?
  • From: Roland Franzius <roland.franzius at uos.de>
  • Date: Sat, 7 Nov 2009 06:46:37 -0500 (EST)
  • References: <hd0t8u$82a$1@smc.vnet.net>

wiso schrieb:
> Look at this:
> 
> Limit[(x^2 - a^2)/(5 x^2 - 4 a x - a^2), x -> a]
> Mathematica answer = 1/3
> 
> this is ok for a !=0, but if a = 0 the value is 
> 
> Limit[(x^2 - a^2)/(5 x^2 - 4 a x - a^2) /. a -> 0, x -> 0]
> 1/5
> 

Simplifying you get as a function of two variables

f: (x,y)->(x+y)/(5x+y)

f is is singular on the line y=-5x. The family of limits f_(x,y)->(0,0) 
depends on the direction it is approching the origin

Limit[f[r Cos[phi],r Sin[phi]],r->0]
=(Cos[phi]+Sin[phi])/(5 Cos[phi]+Sin[phi])

This is a Tan-like function assuming any real value on (0,2 Pi).

For visualization try something like

Plot3D[f[x, y], {x, -0.01, 0.01}, {y, -0.01, 0.015},
  PlotRange -> {-1, 1}, PlotPoints -> 100]

-- 

Roland Franzius



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