For x=a+h, where h>0, then we have two possibilities:
(1) f(x)<0
(2) f(x)>0
In each case |f(x)|>0.
If x=a-h, we also have the same two possibilities, leading to 4 combinations:
(1) f(a+h)>0, f(a-h)>0
(2) f(a+h)>0, f(a-h)<0
(3) f(a+h)<0, f(a-h)>0
(4) f(a+h)<0, f(a-h)<0
In all cases, |f(x)|>0.
If we reduce to, say, h/2, |f(x)|>0. As h→0, |f(x)|→0.
f(x) either approaches zero entirely from the negative or positive side, or f(x) approaches zero from either side, and is “squeezed” between negative and positive.
The left limit as h→0 is zero and the right limit as h→0 is also zero, so the limit of f(x) as x→a (from either side) is zero, because the left and right limits are the same.