Tricky limits question - If $\lim_{(x,y) \rightarrow(0,0)} \ f(0,y)=0$ then $\lim_{(x,y) \rightarrow(0,0)} \ f(x,y)=0$. True or False?

Question

I am asked the following question:

If $\displaystyle \lim_{(x,y) \rightarrow(0,0)} \ f(0,y)=0$ then $\displaystyle \lim_{(x,y) \rightarrow(0,0)} \ f(x,y)=0$

The textbook contains no answer so I would like to check my reasoning on this forum.

Answer

Let's say the question was actually this one:

If $\displaystyle \lim_{(0,y) \rightarrow(0,0)} \ f(x,y)=0$ then $\displaystyle \lim_{(x,y) \rightarrow(0,0)} \ f(x,y)=0$

This, for me, is absolutely incorrect: just because we took one path ($x=0$) and finding a specific value for the limit on that path, doesn't allow us to generalize for the whole limit.

Yes, I know, that's not the original question. And that's where I struggle.

For the original question, I would say it's false but then that would be me just guessing. How should I view the original question?

Answer 1

Let consider as counterexample

$$f(x,y)=\frac{xy}{x^2+y^2}$$

then

• $f(0,y)=0$
• $f(t,t)=\frac12$

Answer 2

It is the same question. Your notation is a bit unusual, perhaps, but what you say is right. In either event, $(x,y)\rightarrow(0,0)$ along the $x-$axis, which doesn't guarantee the existence of the limit.