Show that Dvf(0, 0) does not exist. Find a number α such that if we define f(0, 0) to be equal to α instead of 0, then Dvf(0, 0) will exist.

Question

I only need help with part c

Let $v =\left(\frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}}\right)$. Show that $D_vf(0, 0)$ does not exist. Find a number $\alpha$ such that if we define $f(0, 0)$ to be equal to $\alpha$ instead of $0$, then $D_vf(0, 0)$ will exist.

Is $D_vf(0,0)$ the matrix of partial derivatives evaluated at $(0,0)$ or the directional derivative at $(0,0)$?

Answer 1

$D_vf(0,0)$ is the directional derivative.

Hint: Compute the quotient $\frac{f(tv)-f(0,0)}{t}$

If $f(0,0)=0$ then show that $\lim_{t \to 0}\frac{f(tv)-f(0,0)}{t}$ does not exist.

For the " $ \alpha $"- part: find $ \alpha $ such that $\lim_{t \to 0}\frac{f(tv)-\alpha}{t}$ exists.