If some δ works for a given ε in the limit of f at c, what is always true?

Any smaller δ' also works for that ε

That same δ works for every smaller ε

δ can be chosen independently of ε

Exercise · ZW-LIMIT-OF-A-FUNCTION-020 · ★★★ Advanced

If some \(\delta\) works for a given \(\varepsilon\) in the limit of \(f\) at \(c\), what is always true?

Choose one option.

Show answer

B. Any smaller \(\delta'\) also works for that \(\varepsilon\)

Explanation

If \(0<|x-c|<\delta'\le\delta\), the condition for \(\delta\) already holds. But shrinking \(\varepsilon\) usually requires shrinking \(\delta\): in general \(\delta\) depends on \(\varepsilon\).

Study Calculus in ZWMath →

Review the concept: Limit of a function

The limit of \(f\) as \(x\) approaches \(a\) is \(L\) if \(f(x)\) gets as close to \(L\) as we like when \(x\) is taken sufficiently close to \(a\), regardless of what happens at \(x=a\).

\[\lim_{x\to a} f(x)=L\]

If some \(\delta\) works for a given \(\varepsilon\) in the limit of \(f\) at \(c\), what is always true?

Explanation

Review the concept: Limit of a function

More practice problems: limit of a function