Question

The signum function is defined by \(\operatorname{sgn}(x)=\left\\{\begin{array}{ll}-1, & x<0 \\ 0, & x=0 \\ 1, & x>0\end{array}\right.\) Sketch a graph of \(\operatorname{sgn}(x)\) and find the following (if possible). (a) \(\lim _{x \rightarrow 0^{-}} \operatorname{sgn}(x)\) (b) \(\lim _{x \rightarrow 0^{+}} \operatorname{sgn}(x)\) (c) \(\lim _{x \rightarrow 0} \operatorname{sgn}(x)\)

Short answer

The signum function can be graphed as two horizontal lines at y = -1 and y = 1 with a jump at x = 0. The value of \( \lim_{x \rightarrow 0^{-}} sgn(x)\) is -1. The value of \( \lim_{x \rightarrow 0^{+}} sgn(x)\) is 1. The general limit as \(x\) approaches 0, \( \lim_{x \rightarrow 0} sgn(x)\), does not exist as the limits from left and right are not equal.

Step-by-step solution

Understanding the Signum Function

The signum function, denoted by sgn(x), takes any real number and returns -1 if the number is less than zero, returns 0 if the number is equal to zero, and returns 1 if the number is greater than zero.

Sketching the Graph

Since \(sgn(-x) = -1\) for \(x > 0\), the graph would be a horizontal line along y = -1 for x < 0. For \(x = 0\), \(sgn(x) = 0\), so we can put a mark on the origin. For \(x > 0\), \(sgn(x) = 1\), and, hence, we have another horizontal line along y = 1 for x > 0. Note that there would be a jump at x = 0, as the function instantly jumps from -1 to 1.

Finding the Limit from the Left

The statement \(\lim_{x \rightarrow 0^{-}} sgn(x)\) means that we are looking for the value the function approaches as \(x\) approaches 0 from values less than 0. Based on the graph and definition of the signum function, this value is -1.

Finding the Limit from the Right

The statement \(\lim_{x \rightarrow 0^{+}} sgn(x)\) means that we are looking for the value the function approaches as \(x\) approaches 0 from values greater than 0. Considering the graph and definition of the signum function, this value is 1.

Finding the General Limit

The statement \(\lim_{x \rightarrow 0} sgn(x)\) means we are looking for the value the function approaches as x approaches 0 from either side. If the limits from left and right sides were equal, then we could define \( \lim_{x \rightarrow 0} sgn(x)\). However, the limit from left side is -1, and the limit from right side is 1 which are not equal. Therefore, the general limit as \(x\) approaches 0 does not exist for the signum function.

Key concepts

Sketching Graphs of Functions

Understanding how to sketch a graph of a function is one of the fundamental skills in mathematics. Every function has its own unique graph that visually represents the relationship between its input values (typically along the x-axis) and output values (along the y-axis). The signum function's graph, as shown in the solution provided, is quite distinctive and straightforward to draw. It consists of three separate parts: a horizontal line at y = -1 for all x less than zero, a single point at the origin (0,0), and a horizontal line at y = 1 for all x greater than zero.

This graphical representation is particularly important for visual learners, as it consolidates the abstract relationship between x and y into a concrete form. When dealing with piecewise functions like the signum function, one useful tip is to consider each piece individually, sketch them, and then combine them while paying special attention to transition points, in this case, the origin.

Limits of a Function

Limits are a central concept in calculus, providing a way to describe the behavior of functions as they approach a particular point. For instance, when investigating the limits of the signum function as it approaches zero, two different paths are analyzed: from the left \(x \rightarrow 0^-\) and from the right \(x \rightarrow 0^+\). This approach underscores the notion of one-sided limits, which are particularly relevant when dealing with functions that have a break or jump, as the signum function does at x = 0.

To grasp the concept of limits better, one can imagine tracing the graph of a function from both directions towards a specific point and observing the value it approaches. If these values from left and right differ—as they do for the signum function—the two-sided limit \(\lim_{x \rightarrow 0} sgn(x)\) does not exist, exemplified by the function's discontinuity at x = 0.

Piecewise Functions

Piecewise functions, like the signum function, are defined by multiple sub-functions, each corresponding to a certain interval of the domain. These functions are incredibly versatile, allowing for modeling of situations that aren't consistent across an entire domain but change behavior at certain values.

When dealing with piecewise functions, clarity is crucial: carefully define each sub-function and its interval. In the context of the signum function, we identify three distinct intervals: \(x < 0\), \(x = 0\), and \(x > 0\), with corresponding outputs -1, 0, and 1, respectively. Their explicit boundaries and definitions are essential for correct evaluation, graph sketching, and limit finding. For both students and educators, recognizing the structure of piecewise functions facilitates understanding their properties and predicting their behavior at domain transitions.