Question

What does it mean to say that the limit diverges?

Answer 1

limit can't diverge, sequence, series or function can....it means that don't have an finite value for limit, ''limit'' is infinity

Answer 2

it means that the end result never really wants to settle down to a nice comfy value

Answer 3

In terms of the limit of a sequence we can also consider things that don't become infinity, like the limit of a sequence of a circular trig function like sine of x\[\large\lim_{n \rightarrow \infty}\left\{ \sin x \right\}_{x=0}^{n}\]the sine function oscillates between -1 and 1, but never settles on a particular value, hence the limit of the sequence does not exist, even though infinity never comes up as a value.

Answer 4

Also, a limit like\[\large \lim_{n \rightarrow \infty}\left\{ x\right\}_{x=1}^{\infty}=\infty\]is convergent, because it does approach a particular value, just not a finite one.