I am studying multivariable limits (ML's) in vectorial calculus. My question is at the point 2.
*''does not exist" : DNE
I know when solving a ML whether
- The limit will yield a real value when it is continuous at $\textbf{x}$ ,whether
- The limit does not exist at $\textbf{x}$; and hence you should show there are at least two different values the limit takes when it approaches to $\textbf{x}$ from different paths A, B, C, ... etc.:
So, from a path A the non-existing limit may be a real value p,and from other path B it may produce $\infty$, and from a path C a s $\in\mathbb{R}$ ; my question is:
In order to show the ML DNE, Is it enough to show that the ML is $\infty$ from the path A, and is p$\in\mathbb{R}$ from a path B, and hence we've got two different results? or should the two results given by the differents paths A and B be real numbers, different from $\infty$ ?
In the same way, may I just prove the limit DNE by showing from a path A it produces $\infty$ and from a path B it is $-\infty$?
It is enough to show that you get different limits from two different paths, even if those limits are $+\infty$ or $-\infty$.
The reason is, if $\lim_{\textbf{x}\to\textbf{a}} f(\textbf{x})$ exists and equals $L$ (where $L$ is a real number or $+\infty$ or $-\infty$), then the limit of $f(\textbf{x})$ along any path to $\textbf{a}$ will be $L$. So if two paths give two different values to $L$, then in fact $L$ cannot exist.