iterated limit in 2

Let f be a functionMathworldPlanetmath from a subset S of  2  to  and  (a,b)  an accumulation pointMathworldPlanetmathPlanetmath of S. The limits


are called iterated limits.

Example 1. If  f(x,y):=xsin1x+yx+y,  then

  • limx0(limy0f(x,y))=limx0sin1x does not exist

  • limy0(limx0f(x,y))=limy01=1

  • the usual limit lim(x,y)(0,0)f(x,y) does not exist.

Example 2. If  f(x,y):=x2x2+y2,  then

  • limx0(limy0f(x,y))=limx0x2x2=1

  • limy0(limx0f(x,y))=limy00=0

  • the usual limit lim(x,y)(0,0)f(x,y) again does not exist, though both of the iterated limits do.

So far we have studied examples that present discontinuity at its point of accumulation. We now expose an illustrative example where such discontinuity can be avoided.

Example 3. Consider the function


then (we apply l’Hôpital’s rule ( throughout)

  • limx0(limy0f(x,y))=limx0(limy0xsinxcoshy+ycosxsinhyx2+y2)=limx0xsinxx2=limx0sinxx=limx0cosx=1

  • limy0(limx0f(x,y))=limy0(limx0xsinxcoshy+ycosxsinhyx2+y2)=limy0ysinhyy2=limy0sinhyy=limy0coshy=1

  • the usual limit lim(x,y)(0,0)f(x,y) exists in this case. An essential reason which assures the continuity of this function, arises from the fact that  f(x,y)(sinzz),  z=x+iy,  i.e. it is the real partDlmfPlanetmath of the analytic functionMathworldPlanetmathw:=sinzz  having the removable singularity at  z=0 (see the entry complex sine and cosine).

Title iterated limit in 2
Canonical name IteratedLimitInmathbbR2
Date of creation 2013-03-22 17:28:27
Last modified on 2013-03-22 17:28:27
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Definition
Classification msc 26B12
Classification msc 26A06
Defines iterated limit