f is undefined at x = 1, yet nearby values sit on the line y = x + 1. The hole is not the limit — the height the curve is heading for is.
limx → 1(x² − 1)/(x − 1)=2
The two-sided limit exists and equals 2, even though f(1) does not.
Limit existsThis ε has a δ
Left probe
x = 0.6198
f(x) = 1.6198
|x − a| = 0.3802< δ
|f − L| = 0.3802< ε
Right probe
x = 1.3802
f(x) = 2.3802
|x − a| = 0.3802< δ
|f − L| = 0.3802< ε