Question

What is a limit from below?

Answer 1

If we have a limit from below, that is the same as a limit from the left (more negative).

We can write this like the following:
`lim_(x-> 0^-) f(x)`
rather than the traditional
`lim_(x -> 0) f(x)`

This means we are only considering what happens if we start with a number lower than our limit value and approach it from that direction.

This is generally more interesting with a Piecewise function. Imagine a function which is defined as `y = x` for `x < 0` and `y = x+1` for `x > 0`. We could imagine at that 0 there is a little jump. It should look like this:
graph{|x|/(2x) + 1/2 + x [-3, 3, -2.5, 3.5]}

The limit as `x-> 0` from below is clearly 0 while from above is clearly 1. That means the limit does not exist and there is a jump discontinuity at `x=0`.