Type a formula in x and y (or x, y, z; or add a time t). FMSL computes every value it approaches near the origin, exactly and with proof. Beside it is what the usual "try a few paths" test would conclude.
The real line closed up into a circle: 0 at the bottom, ∞ at the top. The answer is always a point, an arc, or the whole circle.
| along y = 0 | 0 |
| along x = 0 | 0 |
| along y = x | 0 |
| along y = -x | 0 |
| along y = 2x | 0 |
| along y = x^2 | 0 |
| along y = -x^2 | 0 |
| along x = y^2 | 0 |
All paths give 0: the path test concludes the limit is 0.
Which scale y ~ c·|x|^a decides the answer: at each scale the smallest-power terms win.
scaling map (y ~ c|x|^a; size of f ~ |x|^delta): 0 < a < 3 N ~ x^3*y, D ~ y^2: f -> 0 a = 3 BALANCE N ~ x^3*y, D ~ y^2 + x^6: balance: f of order 1, depends on c a > 3 N ~ x^3*y, D ~ x^6: f -> 0 fluid state: [-1/2, 1/2] the end -1/2 is decided on: y = c*|x|^3 (x>0); y = c*|x|^3 (x<0) the end 1/2 is decided on: y = c*|x|^3 (x>0); y = c*|x|^3 (x<0)