Epsilon-delta proof that $\lim_{x \to 2} x^2 = 4$

Primary Tool

Proof Builder

The proof is easier to understand when the bound-setting steps stay in order: choose $δ$ , control $∣ x + 2∣$ , and then estimate $∣ x^{2} - 4∣$ . Seeing those moves line by line makes the logic easier to track.

The key move

The hard part is not the factorization. The hard part is controlling the factor $∣ x + 2∣$ in a way that only depends on a fixed neighborhood around $2$ .

Choosing $δ \leq 1$ gives $1 < x < 3$ , which in turn gives $∣ x + 2∣ < 5$ . That converts the expression into something proportional to $∣ x - 2∣$ alone.

Why the minimum appears

The choice $δ = min (1, ε /5)$ is not decorative. One part keeps $x$ in a region where $∣ x + 2∣$ is bounded, and the other part makes the final estimate smaller than $ε$ .

This is the pattern students should remember: use one bound to control a troublesome factor, and another bound to satisfy the final target.

Common Pitfall

Factoring $x^{2} - 4$ is only the start. You still have to control $∣ x + 2∣$ with a fixed bound, otherwise the estimate does not finish the proof.

Try a Variation

Try adapting the same proof to show $lim_{x \to 1} x^{2} = 1$ . Which constants change, and which part of the argument stays identical?

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ExplanationReal Analysis2D Graphing

What an epsilon-delta proof is actually controlling

An epsilon-delta proof is a control problem: keep $x$ close enough to a point so the function value stays inside a target band around the limit.

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ProofCalculusProof Builder

Proof that a differentiable function is continuous

This proof shows why differentiability is a stronger condition than continuity. The difference $f (a + h) - f (a)$ factors into a difference quotient times $h$ , and that product is forced to zero.

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ExplanationCalculus2D Graphing

Why continuity and differentiability are different concepts

Differentiability is stronger than continuity. A graph can be unbroken at a point and still fail to have a derivative there if the local shape has a corner, cusp, or vertical tangent.