The ε-δ definition pins down what 'limit' means rigorously.
Try this
1.5
f(x) = (x² − 4)/(x − 2), lim (x→2) = 3.5
ε–δ definition
You give me a target tolerance ε; I find a closeness δ that keeps f(x) within ε of L.
Think of it as a challenge game: the skeptic picks ε (how close to L is 'close enough'); you must respond with a δ (how close x must be to a) that always works.
Your turn
For f(x) = 3x, prove lim (x→2) f(x) = 6 by finding a δ for a given ε.
ε-δ
lim(x→a) f(x) = L means: for every ε > 0, there exists δ > 0 such that |x − a| < δ implies |f(x) − L| < ε. Plain English: you can make f(x) arbitrarily close to L by taking x close enough to a.