One-Sided Limits

One-Sided Limits

Consider the function $f(x) = \frac{1}{x}$, and suppose we wanted to evaluate $\lim_{x \to 0} f(x)$:


If we look to the left of the function, it appears as $x \to 0$, $f(x) \to -\infty$. However, if we look to the right of the function, it appears as $x \to 0$, $f(x) \to +\infty$. So then, what is the $\lim_{x \to 0} f(x)$? To answer this question, we define when a limit exists and when it does not exist

Definition: A limit of a function $f$ exists if and only if $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$. If $\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$, then we say $\lim_{x \to a} f(x)$ is undefined.

The notation, "$\lim_{x \to a^-} f(x)$" signifies a lefthand limit. In our example above, $\lim_{x \to 0^-} f(x) = -\infty$.

The notation, "$\lim_{x \to a^+} f(x)$" signifies a righthand limit. In our example above once again, $\lim_{x \to 0^+} f(x) = + \infty$.

By our definition, since $\lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x)$, then $\lim_{x \to 0} f(x)$ is undefined or does not exist (often abbreviated D.N.E.).

Example 1

Evaluate $\lim_{x \to 4^-} \left ( \frac{x}{x - 4} + 2 \right )$.

The following is a graph of the function $f(x) = \frac{x}{x - 4} + 2$:


From our graph we clearly see that $\lim_{x \to 4^-} f(x) = -\infty$.

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