Hey everyone. I'm having some conflict with Squeeze Theorem for two variable functions. Here are a couple of examples:

Now one can say,

Therefore, limit exists. Or does it?

If we let and approach (0,0) along the line y = mx, this happens:

∴

does not exist because one gets a different answer for all non vertical line. Thus, limit does not exist. But then why did Squeeze Theorem say it did?

Another example:

Now one can say,

Then clearly

as (x,y) approaches (0,0). So by the squeeze theorem,

as (x, y) approaches (0, 0)

Why does the latter example hold true while the former does not? Thank you in advance!