Order the steps in an epsilon-delta proof that lim x→ a f(x) = L.

Let ε > 0 be given (arbitrarily small) → Express |f(x) − L| in terms of |x − a| → Choose δ > 0 in terms of ε to control |f(x) − L| → Verify: |x − a| < δ implies |f(x) − L| < ε

Physics I / Limits and continuity

Practice question · Put in order

Order the steps in an epsilon-delta proof that $lim_{x\to a} f(x)$ = L.

Choose $\delta$ > 0 in terms of $\varepsilon$ to control | $f(x)$ − L|
Express | $f(x)$ − L| in terms of |x − a|
Let $\varepsilon$ > 0 be given (arbitrarily small)
Verify: |x − a| < $\delta$ implies | $f(x)$ − L| < $\varepsilon$

Hints

The challenge quantity is named first; your response to it comes second.
Delta is chosen in terms of epsilon, so it cannot come first.

Show the answer

Let $\varepsilon$ > 0 be given (arbitrarily small)
Express | $f(x)$ − L| in terms of |x − a|
Choose $\delta$ > 0 in terms of $\varepsilon$ to control | $f(x)$ − L|
Verify: |x − a| < $\delta$ implies | $f(x)$ − L| < $\varepsilon$

Why

The epsilon-delta definition formalises the intuitive notion of a limit.

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Order the steps in an epsilon-delta proof that limx→af(x)lim_{x\to a} f(x)limx→a​f(x) = L.

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Order the steps in an epsilon-delta proof that $lim_{x\to a} f(x)$ = L.