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The precise definition of the left-hand limit, $\lim _{x \rightarrow a^{-}} f(x)=L$, may be stated as follows: For every number $\varepsilon>0$ there exists a number $\delta>0$ such that $|f(x)-L|<\varepsilon$ whenever $a-\delta<x<a$. Similarly, for the right-hand limit, $\lim _{x \rightarrow a^{+}} f(x)=L$ if for every number $\varepsilon>0$ there exists a number $\delta>0$ such that $|f(x)-L|<\varepsilon$ whenever $a<x<a+\delta$. Explain, with the aid of figures, why these definitions are appropriate.

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