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Question on theory: If $\lim_{x\to a^+}F(x)=L$ and $\lim_{x\to a^-}F(x)$ doesn't exist, then does the $\lim_{x\to a}F(x)$ exists and is $L$?

Thanks.

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What is the exact definition of the limit of a function at a point? Isn't the right limit and the left one should equal? –  Babak S. Dec 24 '12 at 15:09

3 Answers 3

It depends on what you mean $\lim_{x\to a^-}F(x)$ doesn't exist.

If $F$ is not defined in $(a-\delta,a)$ for no $\delta>0$ then $x$ can't approach $a$ from the left and $\lim_{x\to a^-}F(x)$ is not defined. If that's the case then, $\lim_{x\to a}F(x)$ exists and is $L$

If $F$ is defined in $(a-\delta,a)$ for some $\delta>0$ but $\lim_{x\to a^-}F(x)$ does not exist then $\lim_{x\to a}F(x)$ does not exist

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No, if the limit exists, then it must exist from both sides, and those two directional limits must be equal.

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If the function $F(x)$ has domain say $[a,b]$ where $b>a$, so that $F(x)$ is not even defined for $x<a$, and if $\lim_{x \to a^+}F(x)=L$, then using some definitions of limit, the limit exists. Sometimes limit is defined to mean that, if $x$ approaches $a$ such that $x$ stays in the domain of $F(x)$, then the limit exists.

But many calculus texts insist on limits being defined on both sides of $a$ for the limit to be said to exist, and such texts would say only that the "one sided" limit exists.

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