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I know the definition of the limit of functions

Let $A \subseteq \mathbb{R}$ and let $f: A \rightarrow \mathbb{R}$. We say $$\lim \limits_{x \to a} f(x) = \mathscr l$$ if $(\forall \epsilon > 0)(\exists \delta > 0)(\forall x \in A), 0 < |x-a| < \delta \Rightarrow |f(x) - \mathscr l| < \epsilon$.

My question is can we say :

Let $A \subseteq \mathbb{R}$ and let $f: A \rightarrow \mathbb{R}$. If $$\lim \limits_{x \to a} f(x) = \mathscr l$$ then $(\forall \epsilon > 0)(\exists \delta > 0)(\forall x \in A), 0 < |x-a| < \delta \Rightarrow |f(x) - \mathscr l| < \epsilon$.

Note: The question I was working on has a well-defined limit.

Many thanks

Thomas Andrews
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MCJ
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1 Answers1

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Although usually phrased using the "if..." clause in English, mathematical definitions are "if and only if" conditions.

Once the limit notation is defined, yes, your "if... then..." statement is true.