$f \in c^\infty: f(x, y) = 0$ only at the origin. What happens with the gradient?

Question

I have a function $f \in c^n$ such that:

• $f(0, 0) = 0$
• $f(x,y) \neq 0 (\forall (x,y) \neq (0,0))$

I want to prove that the parcial derivatives at the origin are both 0. The thing is that I'm stuck with these limits:

• $\lim_{t \to 0} \frac{f(t,0)}{t}$
• $\lim_{t \to 0} \frac{f(0,t)}{t}$

How can I solve those indeterminations?

It is valid to state that since $f$ has continuous derivatives, both can't go to infinity and therefore those limits go to 0?