Epsilon-Delta: The Definition of a Limit
$\lim_{x \to a} f(x) = L$ means: for every $\varepsilon > 0$ there is a $\delta > 0$ such that $|f(x) - L| < \varepsilon$ for every $x$ with $0 < |x - a| < \delta$. The horizontal band is the $\varepsilon$-neighbourhood of $L$, the vertical band the $\delta$-neighbourhood of $a$. If the piece of the curve inside the vertical band stays inside the horizontal band, your $\delta$ is good enough for that $\varepsilon$.
What to notice
- The order in the definition matters: $\varepsilon$ is given first, then a $\delta$ is sought for it. Making $\varepsilon$ smaller usually calls for a smaller $\delta$.
- If one $\delta$ works, every smaller $\delta$ works too; you do not have to find the largest.
- In the examples with a hole the function is undefined at $a$, yet the limit exists: the definition never looks at $x = a$ itself ($0 < |x - a|$).
- In the jump example bring $\varepsilon$ below $1$: no $\delta$ is enough. If no $\delta$ can be found for even a single $\varepsilon$, the limit is not that number.