Problem 116 - Calculus of a Single Variable

Question

Signum Function 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). $$ \begin{array}{lll}{\text { (a) } \lim _{x \rightarrow 0^{-}} \operatorname{sgn}(x)} & {\text { (b) } \lim _{x \rightarrow 0^{+}} \operatorname{sgn}(x)} & {\text { (c) } \lim _{x \rightarrow 0} \operatorname{sgn}(x)}\end{array} $$

Step-by-Step Solution

Answer

The left-hand limit of sgn(x) as x approaches 0 is -1. The right-hand limit of sgn(x) as x approaches 0 is 1. The limit of sgn(x) at x = 0 does not exist.

Step 1: Graphing the Signum Function

Start by drawing the graph according to the definition of the signum function. The function equals -1 when x < 0, equals 0 when x = 0, and equals 1 when x > 0. The x-axis will be divided into 3 segments by the vertical lines x = 0, with the segment to the left of x=0 comprising of y=-1, at x=0 y=0 and for x>0, y=1. The resulting graph therefore consists of two horizontal lines and a point at the origin (0, 0).

Step 2: Determining the left-hand limit

The left-hand limit of a function as x approaches a certain value is the value that the function approaches as x gets infinitely close to that value from the left. In the case of the signum function, as x approaches 0 from the left, sgn(x) approaches -1, because for all x<0, sgn(x) = -1.

Step 3: Determining the right-hand limit

The right-hand limit is similarly defined, only that x gets infinitely close to the value from the right. As x approaches 0 from the right, sgn(x) approaches 1, because for all x>0, sgn(x) = 1.

Step 4: Determining the limit at the point

The limit of a function at a certain point is the value that the function approaches as x gets infinitely close to that value from both the left and the right. However, in this case, the function does not approach a single value, as the left-hand and right-hand limits are not equal. Hence, the limit of sgn(x) at x = 0 does not exist.

Key Concepts

Understanding Piecewise Functions

Piecewise functions are fascinating as they are defined by multiple sub-functions, each applicable to a different interval of the domain. In the case of the Signum function, the function is defined as:

• -1 when \(x < 0\)
• 0 when \(x = 0\)
• 1 when \(x > 0\)

This means the domain is split into segments, with a specific value assigned for each segment.
These types of functions are quite common and crucial in mathematics, especially when modeling situations that have different conditions or rules for different scenarios. It's like a traffic light: it changes according to different rules for various situations.
By understanding piecewise functions, students can better grasp situations in calculus where behavior changes abruptly, ensuring a deeper knowledge of how mathematical functions can model real-world behavior.

Exploring Limits in Functions

Limits play a crucial role in calculus, representing the value that a function approaches as the input gets closer to a certain point. For the signum function, limits help determine behavior as \(x\) approaches zero from the left or right.
In evaluating limits, we are essentially asking: "As we edge closer and closer to a particular value, what does the function tend towards?"
For example:

• The left-hand limit \(\lim\limits_{x \to 0^-} \operatorname{sgn}(x) = -1\), because as \(x\) approaches zero from the left, the function consistently outputs -1.
• The right-hand limit \(\lim\limits_{x \to 0^+} \operatorname{sgn}(x) = 1\), as approaching zero from the right yields a value of 1.

Limits help analyze the trend, and not just the particular values a function reaches. Through limits, students understand how a function behaves "infinitely closely" to a point, which is fundamental in calculus concepts like continuity and derivatives.

Graphing Functions like Signum

Graphing functions is a vital skill in visualizing their behavior and understanding how different inputs affect the outputs. For the Signum function, graphing involves drawing straightforward horizontal lines based on the piecewise definition.
The steps to graph the signum function are:

• Draw a horizontal line at \(y = -1\) for \(x < 0\).
• Mark a point at the origin where \(x = 0\) and \(y = 0\).
• Draw another horizontal line at \(y = 1\) for \(x > 0\).

Graphing these horizontal lines highlights the distinct sections that make up piecewise functions. The visual appearance will consist of two horizontal lines and a singular point at the origin, helping students recognize the discrete nature of piecewise-defined functions.
Graphs provide a visual means to comprehend concepts, presenting an illustration of how the function behaves across different intervals, significant especially in studying behaviors like continuity and discontinuity.

Discontinuity in Functions

Discontinuity in functions occurs when a function is not smooth or uninterrupted at a point or over an interval. For the signum function, this happens at \(x = 0\), where there is a jump from -1 to 1.
Discontinuity is seen at any point where the left-hand and right-hand limits are unequal, meaning the function behaves differently as it approaches from opposite directions.
Understanding discontinuity involves:

• Recognizing where a function's values change abruptly.
• Using limits to assess the behavior of the function as it nears the point in question.
• Realizing that a function with discontinuity doesn't have a limit at the point; for \(\operatorname{sgn}(x)\), \(\lim\limits_{x \to 0} \operatorname{sgn}(x)\) does not exist, because the left and right limits are different.

Discontinuity is an important feature when analyzing functions, as it influences the continuity and limits-related properties, which are fundamental in the study of calculus and numerical approximations.