University Calculus: Early Transcendentals (3rd Edition)

Take these three limits: $\lim_{x\to c}f(x)$, $\lim_{x\to c^+}f(x)$ and $\lim_{x\to c^-}f(x)$ While limits are the values that a function $f(x)$ approaches as $x$ approaches a number $c$, one-sided limits are the values that a function $f(x)$ approaches as $x$ approaches a number $c$ from only one side, either the left one or the right one. So we can think of limits as two-sided ones, as we consider $x$ approaches $c$ from both its left and right sides. This means that $\lim_{x\to c}f(x)$, which is the limit, only exists if both of the following conditions are satisfied: - Both $\lim_{x\to c^+}f(x)$ and $\lim_{x\to c^-}f(x)$ exist - $\lim_{x\to c^+}f(x)=\lim_{x\to c^-}f(x)$ because the limit is two-sided, for limit to exist, $f(x)$ must approach a single value from both two sides. - Example: We have a function $f(x)=2$ for $x\lt2$ and $f(x)=5$ for $x\ge2$. Here $\lim_{x\to 2^-}f(x)=2$ and $\lim_{x\to 2^+}f(x)=5$. Since they are not equal, $\lim_{x\to 2}f(x)$ does not exist. If we draw a graph, we will see $f(x)$ values jump at $x=2$. On the other hand, if we know for sure that $\lim_{x\to c}f(x)$ exists, then calculating either $\lim_{x\to c^+}f(x)$ or $\lim_{x\to c^-}f(x)$ is enough to conclude the value of $\lim_{x\to c}f(x)$. This is because $f(x)$ must approach the same value when $x\to c$ as when $x\to c$ from the left or the right so that $\lim_{x\to c}f(x)$ can exist. In other words, $\lim_{x\to c}f(x)=\lim_{x\to c^+}f(x)=\lim_{x\to c^-}f(x)$ - Example: $f(x)=x^3-1$ We know that $\lim_{x\to 3}f(x)$ exists. Calculating $\lim_{x\to 3^+}f(x)=3^3-1=26$ We can conclude that $\lim_{x\to 3}f(x)=26$