# Chapter 2 - Section 2.4 - One-Sided Limits - Exercises - Page 85: 43

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 $a$.

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