Answer
If we know $\lim_{x\to a^+}f(x)$ and $\lim_{x\to a^-}f(x)$ at an interior point of the domain of $f$, we will know $\lim_{x\to a}f(x)$.
Work Step by Step
If we know $\lim_{x\to a^+}f(x)$ and $\lim_{x\to a^-}f(x)$ at an interior point of the domain of $f$, we will know $\lim_{x\to a}f(x)$.
In fact, there will be 3 cases:
1) Case 1: $\lim_{x\to a^+}f(x)=\lim_{x\to a^-}f(x)= X$
In this case, $\lim_{x\to a}f(x)=X$, because as $x$ approaches $a$ from both the left and right side of $a$, $f(x)$ would approach one and only one value, which is $X$. So $\lim_{x\to a}f(x)=X$ whenever $\lim_{x\to a^+}f(x)=\lim_{x\to a^-}f(x)=X$
2) Case 2: $\lim_{x\to a^+}f(x)\ne\lim_{x\to a^+}f(x)$
In this case, $\lim_{x\to a}f(x)$ does not exist, because there are 2 different values that $f(x)$ approaches as $x$ approaches $a$ from the left and from the right. In brief, $f(x)$ does not approach any single value as $x$ approaches $a$, so $\lim_{x\to a}f(x)$ does not exist.
3) Case 3: $\lim_{x\to a^+}f(x)$ or $\lim_{x\to a^-}f(x)$ does not exist or both of them do not exist
In this case, $\lim_{x\to a}f(x)$ also does not exist, because to be counted that $f(x)$ approaches a value $X$ as $x$ approaches $a$, $f(x)$ must approach $X$ as $x$ approaches $a$ from both the left and right side of $a$.
