Limits Existence Implies Equality Along All Paths

If ${\lim }_{x\to x_0}f(x)$ exists, then it agrees along all path

Theorem

If $c(t)$ is a path with ${\lim }_{t\to t_0}c(t)=x_0$ Then, ${\lim }_{x\to x_0}f(x)={\lim }_{t\to t_0}f(c(t))$

Also helpful to show that limits do not exist, if a limit does not agree along any two paths.