$∀ε>0,∃δ>0 $ s.t. $0< |x-a|<δ⇒|f(x)-L|<ε$
In the part where it says: "$0< |x-a|<δ$", what is the purpose of "$0<$"?
What if it was not there? How would that change the definition?
2026-04-05 16:14:13.1775405653
What is the significance of the "0<" in the definition of limits?
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If you allow $x$ to be equal to $a$ in the definition of a limit, it would imply every function defined at $a$ which has a limit would be continuous at $a$.
Thus, for instance, the function defined as $\smash{\begin{cases} f(x)=x&\text{if }\; x\ne 0,\\ f(0)=1,\end{cases}}\;$ would have no limit when $x$ tends to $0$, which is absurd.