This is the negation of Cauchy's definition of a limit of a function
It is used to prove a limit does not equal L (any given number)
See Link for Cauchy's definition of a limit.
~the negative of "for all" is "there exists"
~the negative of "there exists " is "for all"
~such that remains the same
~the negative of if P, then Q is found this way:
P implies Q means not P or Q in logic: (~P or Q)
so an if, then statement is equivalent to an OR statement.
The negative of an OR statement is an AND statement.
DeMorgan's Law in logic gives this: ~(~P OR Q) is equivalent
to ~(~P) AND ~Q or P AND ~Q
So, the negative of Cauchy's definition becomes:
There exists e>0, such that, for all d>0, the following
must hold: 0< |x-a| < d AND |f(x)-L| ≥ e
Which means that we have to find just one e-neighborhood of L,
such that, for all d's (for all neighborhoods of a), there will be x's in
those neighborhoods that produce f(x)'s outside of that e-neighborhood
of L. This will prove that the the limit is not L.
x+1, if x≥ 0
Example: Take the function f(x)= {
x-1, if x < 0
the graph of f has a jump discontinuity at 0.
The limit cannot be 1, as x → 0, since taking e small enough, (say 0.1), every
neighborhood of 0 would give f(x) values that are negative (from the
left side of 0), that are not in that e-neighborhood of 1. For a similar
reason, the limit cannot be -1, since values of x on the right side of 0
give f(x) that are positive. No matter what limit you assume, you will
always be able to find a neighborhood size e about L that will not
contain values produced by at least some x's in any d-neighborhood of 0.
This will prove the limit is not whatever number you choose.