One-Sided Limits and Infinite Limits

Tags: #◇Computer Science #◇CSMATH1 #◇Problem Solving

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◇Overview

One-Sided Limits

A one-sided limit restricts the approach of $x$ to a specific side (either left or right) of the value $c$. Left-hand limit: The value $f(x)$ approaches as $x$ gets closer to $c$ from values less than $c$. Denoted as $\lim_{x \to c^-} f(x) = L$. Right-hand limit: The value $f(x)$ approaches as $x$ gets closer to $c$ from values greater than $c$. Denoted as $\lim_{x \to c^+} f(x) = L$.
A two-sided limit $\lim_{x \to c} f(x)$ exists if and only if both the left-hand and right-hand limits exist and are equal.

Infinite Limits

An infinite limit occurs when the function values increase or decrease without bound as $x$ approaches $c$. If $f(x)$ increases without bound, we write $\lim_{x \to c} f(x) = +\infty$. If $f(x)$ decreases without bound, we write $\lim_{x \to c} f(x) = -\infty$.

Infinite limits typically indicate the presence of a vertical asymptote at $x=c$. If $f(x)$ is a rational function $\frac{p(x)}{q(x)}$ where $q(c)=0$ and $p(c) \neq 0$, the function will have an infinite limit at $c$.

◇Important Formulas

Right-Hand Limit: $\lim_{x \to c^+} f(x)$ Left-Hand Limit: $\lim_{x \to c^-} f(x)$ Vertical Asymptote: The line $x = c$ is a vertical asymptote of $f$ if at least one of the following is true:
$\lim_{x \to c^-} f(x) = \pm \infty$
$\lim_{x \to c^+} f(x) = \pm \infty$

◇Worked Examples

Example 1: Left and Right-Hand Limits
Determine the one-sided limits of $f(x) = \frac{|x - 2|}{x - 2}$ as $x \to 2$.
Solution:
For $x > 2$, $|x - 2| = x - 2$, so $f(x) = \frac{x - 2}{x - 2} = 1$. Thus, $\lim_{x \to 2^+} f(x) = 1$.
For $x < 2$, $|x - 2| = -(x - 2)$, so $f(x) = \frac{-(x - 2)}{x - 2} = -1$. Thus, $\lim_{x \to 2^-} f(x) = -1$.

Example 2: Infinite Limits
Investigate the limit of $f(x) = \frac{1}{x^2}$ as $x \to 0$.
Solution:
As $x$ approaches $0$ from either the left or the right, $x^2$ becomes a very small positive number.
Therefore, $\frac{1}{x^2}$ becomes an arbitrarily large positive number.
$\lim_{x \to 0^-} \frac{1}{x^2} = +\infty$
$\lim_{x \to 0^+} \frac{1}{x^2} = +\infty$
Thus, $\lim_{x \to 0} \frac{1}{x^2} = +\infty$. The line $x=0$ (the y-axis) is a vertical asymptote.

◇References & Navigation

• Previous Topic: ◇Definition of the Limit and Limit Theorems
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• Related Notes: ◇Introduction to Limits, ◇Limit at Infinity