If the Limit Does Not Exist Does That Imply Divergence?

Question

Let say $$\lim_{x\to p}f(x)=q$$

Think about two cases:

1. $p$ is not the limit point of the function domain $E$. Does this limit exist? diverge?
2. $p$ is the limit point of the function domain $E$, but we cannot find such a $q$. Does this limit exist? diverge?

I am just confused that is a limit does not exist equal to diverge?

Answer 1

There are several ways in which a limit may fail to exist:

• the function runs of to $\pm \infty$, like $\lim_{x\rightarrow\infty}x$

• the function remains finite but keeps oscillating, like $\lim_{x\rightarrow 0} \sin(1/x)$

• the left limit does not equal the right limit, like $\lim_{x\rightarrow 0} \frac{|x|}{x}$

I would only call the first of these "diverging".