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Mathematica 2D limit computation

Wolfram Alpha and Mathematica (on my laptop) give zero for the limit shown in the image below.

This is okay if x and y approach the origin along the path y = x.

But what happens if x and y approach the origin along the path y = x^3?

I have been unable to find any Stack Overflow questions that address this issue.

Wolfram Alpha limit computation

Solution

  • Limits of a function f along path p depends on which path is taken. You hint at this in the question. If we insert y=x^3 into f, we get the constant 1/2. So the limit of f towards (0,0) along the path y=x^3 is 1/2.

    Mathematica only computes limits along a one axis at a time. Even thought WolframAlpha makes it look like it knows how to compute the correct (x,y)->(0,0) it actually computes lim x->0 lim y->0 f(x,y).

    This question and answer can be used to examine the situation graphically: https://mathematica.stackexchange.com/a/21549/11860