Question

What does limit does not exist mean?

Answer 1

For the definition, please see below.

Explanation:

`lim_(xrarra)f(x)` does not exist
The idea is that there is no number that `f(x)` gets arbitrarily close to for `x` sufficiently close to `a`.

For a function `f` defined in some open interval that contains `a`, except possibly at `a`,

`lim_(xrarra)f(x)` does not exist if and only if

there is no number `L` such that for every `epsilon > 0`, there is a `delta > 0` with: for all `x`, if `0 < abs(x-a) < delta`, then `abs(f(x)-L) < epsilon`

equivalently

for every `L` there is some `epsilon > 0` such that for every `delta > 0` there is some `x` with `0 < abs(x-a) < delta`, and `abs(f(x)-L) >= epsilon`