If lim x → a f(x) = L then f(a) = L.

Practice question · True or false

If $\lim_{x \to a} f(x) = L$ then $f(a) = L$ .

Hints

Does the limit inspect the value at $a$ ?
A removable discontinuity has both, and they disagree.

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False

Why

False. The limit describes behaviour approaching $a$ and deliberately ignores the value there, $f(a)$ may differ from $L$ , or not exist at all. The two coinciding is precisely the definition of continuity at $a$ , which is why continuity has to be stated as a separate condition rather than following from the limit.

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If lim⁡x→af(x)=L\lim_{x \to a} f(x) = Llimx→a​f(x)=L then f(a)=Lf(a) = Lf(a)=L.

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If $\lim_{x \to a} f(x) = L$ then $f(a) = L$ .