How to find slope at a point where the derivative is indeterminate

Question

How should I find the slope of a curve at origin whose derivative at the origin is indeterminate. My original problem is to calculate the equation of tangent to a curve at origin. But for the equation we need to get the slope but the derivative at the origin is indeterminate. For eg $$ x^3+y^3 =3axy $$ which though has $x=0$ zero and $y=0$ as tangents yet its derivative at origin is indeterminate. Note- this is only one example there may be other examples where the derivative of the curve at origin is indeterminate but the curve has a tangent there at the origin and moreover the tangent need not be the coordinate axes. So it is sure that even if the derivative is indeterminate at origin yet the curve can have a tangent with a slope other than infinite or zero. But how to find the slope

Answer 1

As far as I can tell from your example the problem is having more than one tangent at hand to assign to the origin.

The derivative has to be unique, so I do not think this can be fixed.

Answer 2

If the derivative is indeterminate you might try l'Hospital rule to reduce the expression to a determinate form.
And if you mean infinite instead, that would mean the tangent line is vertical (parallel to the ordinate axis).