Let f be defined as:
$$f(x,y)=\frac{\sin(x^3+y^3)}{x^2+y^2}$$
and $f(0,0)=0$.
a) For which directions $v \in \mathbb{R}^2\setminus\{0\}$ is there a directional derivative $d_vf(0,0)$. Calculate it in these cases.
b) Is $f$ totally differentiable at the origin?
My approach: find the partial derivatives over $x$ and $y$ in the origin. If they exist and are continuous, then $f$ is totally differentiable at the origin (answering b) and also we can calculate the directional derivative for any $v$ as
$$\frac{df}{dv}(0,0) = f'(x)v$$
My first question: will this approach work? Secondly: I'm stuck calculating the partial derivatives:
$$\lim_{h\rightarrow \infty} \frac{\sin(h^3)}{h^2}$$
How do I prove what that limit is?
Best Answer
You can compute $\frac{\partial f}{\partial x}(0,0)$ by the definition:$$\frac{\partial f}{\partial x}(0,0)=\lim_{h\to0}\frac{f(h,0)-f(0,0)}h=\lim_{h\to0}\frac{\sin(h^3)}{h^3}=1$$and, by the same argument, $\frac{\partial f}{\partial y}(0,0)=1$. So, if $f$ is differentiable at $(0,0)$, you must have$$f'(0,0)(a,b)=\frac{\partial f}{\partial x}(0,0)a+\frac{\partial f}{\partial y}(0,0)b=a+b.$$
But $f$ is not differentiable at $(0,0)$, since this would mean that$$\lim_{(x,y)\to(0,0)}\frac{\bigl\lvert f(x,y)-f(0,0)-f'(0,0)(x,y)\bigr\rvert}{\lVert(x,y)\rVert}=0,$$which is equivalent to$$\lim_{(x,y)\to(0,0)}\frac{\bigl\lvert\sin(x^3+y^3)-x-y\bigr\rvert}{(x^2+y^2)^{3/2}}=0.$$However, this is not true; just see what happens if $y=x$.