Given the following function

$$f(x,y)=\begin{cases}\frac{x^{707}y}{x^{1414}+y^2}, &(x,y) \neq (0,0)\\0 ,&(x,y)=(0,0) \end{cases}$$

Does $f$ have a directional derivative in every direction at $(0,0).$

What i tried

Using the definition of the directional derivative, $$ \lim_{t \to 0} \frac{f(a + tv) - f(a)}{t}$$

And letting $(a,b)$ be the unit vector, i plug $f(0,0)$ and $(a,b)$ into the defintion of the directional derivative, hence getting after simplification $$\lim_{t \to 0} \frac{a^{707}bt^{705} }{a^{1414}t^{1412}+b^2}$$ and i mentioned that for $b$ not equals $0$, the limit tends to $0$, while for $b$ equals $0$ the limit also equals $0$, hence the directional derivative does not exists at $(0,0)$.For a directional derivative to exists, the limit cannot be $0$.Am i correct, could anyone explain. Thanks


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