I will explain my question with the help of an example. We need to find the tangent at origin to the curve $$x^3 + y^3 =3axy$$
The derivative at origin is $0/0$ or indeterminate, found after implicit differentiation. But the tangents exist (via Wolfram Alpha) and they are $x=y=0$.
- If the derivative at the origin does not exist, how are we getting the tangents? At least $y=0$ has a determinate slope (0).
- Also how should I find tangents to more general curves at points where the derivative doesn't exist? Is there a general method using differentiation?
My professor told me that as $x,y\to0$, $x^3 + y^3\ll3axy$ and hence the zeroes of the function will be approximately where the zeroes of $3axy$ are. Now I couldn't understand the next line that he said:
Near the origin the curve will look like the solutions to $3axy$.
What does he mean by this? Of course the solutions to $3axy=0$ are $x=0$ and $y=0$, which are the tangents, but the curve isn't like that.
Can anyone please explain me this? And is there a general method to find tangents at points where the derivative doesn't exist?