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I have the following definition:

Let $I\subseteq \Bbb R$ be an open interval and let $c\in I$. A function $f:I \setminus \{c\}\rightarrow \Bbb R$ has a left hand limit in the point $c$ equal to $L$ if:

$$(\forall \varepsilon>0)(\exists \delta>0)(\forall x\in I)((0<c-x<\delta)\Rightarrow(|f(x)-L|<\varepsilon)).$$ Then we say that $\lim_{x\to c^-}f(x)=L.$

Can someone please explain this definition with an example or tell me where I can find some help on understanding this definition?

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    $\begingroup$ Informally it means: no matter what size interval around L you give me (on the y axis), I can find an interval on the x axis of the form (c- something ositive, c) such that ALL the points from that interval map to something within the interval around L you gave me. $\endgroup$ – Ovi Feb 8 '17 at 17:21
  • $\begingroup$ $c\in I^\circ$. $\endgroup$ – Nosrati Feb 8 '17 at 17:24
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Draw the graph of $$f(x) = \left\{ \array{x & (x < 1) \\ x+1 & (x\ge 1)} \right.$$ The left hand limit of $f$ at $x=1 $ is $=1$, the right hand limit (defined similarly) is $=2$

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