Why does the limit not exist?

Question

I'm working on partial derivatives and I'm taking calculus IV. I need some help understanding why part $ c$'s limit "does not exist".

Find each of the following limits, or explain that the limit does not exist.

Let $f(x,y) =\begin{cases} 1&, y \geq x^4\\ 1&, y \leq 0 \\ 0&, \text{otherwise} \end{cases}$

a) limit of $f(x,y)$ as $(x,y)$ approaches $(0,1)$;

b) limit of $f(x,y)$ as $(x,y)$ approaches, $(2,3)$;

c) limit of $f(x,y)$ as $(x,y)$ approaches $(0,0)$.

Is part $c$ "does not exist" because the limits of part a and b are different? Or am I wrong? Thanks.

Answer 1

If you approach $(0,0)$ along the path $x = 0$, then we get a limit of $1$, since we always fall under either the first or the second cases of the piecewise function.

If you approach $(0,0)$ along the path $y = 0.5x^4$, then we get a limit of $0$, since we always fall under the third case of the piecewise function.

Since different paths yielded different limits, it follows that: $$ \lim_{(x,y) \to (0,0)} f(x,y) $$ does not exist.

Answer 2

For a limit to exist, it must be independent of the path of approach. Note that this is a necessary but not a sufficient condition for existence.

Approaching the origin along $y=x^4$ and along the path $y = ky^4 $ where $0<k<1$ yield different limits (try it), and so this particular limit cannot exist.