Problem 47 - Calculus: Early Transcendentals

Question

The \( signum \) (or sign) \( function\) , denoted by sgn, is defined by sgn \( x = \left\\{ \begin{array}{ll} -1 & \mbox{if \) x < 0 \(}\\\ 0 & \mbox{if \) x = 0 \(}\\\ 1 & \mbox{if \) x > 0 \(} \end{array} \right.\) (a) Sketch the graph of this function. (b) Find each of the following limits or explain why it does not exist. (i) \( \displaystyle \lim_{x \to 0^+}\text{sgn \) x \(} \) (ii) \( \displaystyle \lim_{x \to 0^-}\text{sgn \) x \(} \) (iii) \( \displaystyle \lim_{x \to 0}\text{sgn \) x \(} \) (iv) \( \displaystyle \lim_{x \to 0}| \text{sgn \) x \(} | \)

Step-by-Step Solution

Answer

Graph: step-wise with lines at y=-1, 0, 1. Limits: (i) 1, (ii) -1, (iii) DNE, (iv) 1.

Step 1: Understand the signum function

The signum function, denoted as \( \text{sgn}(x) \), categorizes the number based on its sign. If \( x < 0 \), the function outputs \(-1\). If \( x = 0 \), the output is \(0\). Lastly, if \( x > 0 \), the function outputs \(1\). This means it only checks if a number is negative, zero, or positive and outputs based on these conditions.

Step 2: Sketch the graph of sgn(x)

On the y-axis, the function \( \text{sgn}(x) \) has values of \(-1\) for \( x < 0 \), \(0\) at \( x = 0 \), and \(1\) for \( x > 0 \). On a graph, this translates to a horizontal line at \( y = -1 \) for negative \( x \) values, a point at \((0, 0)\), and a horizontal line at \( y = 1 \) for positive \( x \) values. There will be open and closed points at zero to indicate that \( \text{sgn}(x) = 0 \) exactly at zero.

Step 3: Evaluate the limit as x approaches 0 from the positive side

For \( \lim_{x \to 0^+} \text{sgn}(x) \), as \( x \) approaches zero from the positive side, the value of \( x \) stays positive, so \( \text{sgn}(x) \) approaches \(1\).

Step 4: Evaluate the limit as x approaches 0 from the negative side

For \( \lim_{x \to 0^-} \text{sgn}(x) \), as \( x \) approaches zero from the negative side, the value of \( x \) stays negative, so \( \text{sgn}(x) \) approaches \(-1\).

Step 5: Evaluate the two-sided limit at x equals 0

\( \lim_{x \to 0} \text{sgn}(x) \) does not exist because the left-hand limit (\(-1\)) and the right-hand limit (\(1\)) are not equal. For this limit to exist, the left-hand and right-hand limits must agree, which is not the case here.

Step 6: Evaluate the limit of the absolute value of sgn(x) as x approaches 0

The expression \(|\text{sgn}(x)|\) takes the absolute value of the signum function, so it outputs \(1\) for both negative and positive values of \(x\) since both \(-1\) and \(1\) have an absolute value equal to \(1\). Thus, both \( \lim_{x \to 0^+}|\text{sgn}(x)| \) and \( \lim_{x \to 0^-}|\text{sgn}(x)| \) equal \(1\). Therefore, \( \lim_{x \to 0}|\text{sgn}(x)| = 1 \) because the one-sided limits agree.

Key Concepts

Limits

Limits describe the behavior of a function as the input, or x-value, approaches a particular point. For instance, when discussing the limit of the signum function, \( \text{sgn}(x) \), as \( x \to 0 \), we're interested in what the function approaches as x gets infinitely close to zero from either the positive or negative side.

• If \( x \) approaches 0 from positive values, \( \text{sgn}(x) \) approaches 1.
• If \( x \) approaches 0 from negative values, \( \text{sgn}(x) \) approaches -1.

These are known as one-sided limits, \( \lim_{x \to 0^+} \) and \( \lim_{x \to 0^-} \) respectively. Since these two one-sided limits do not agree, the two-sided limit \( \lim_{x \to 0} \text{sgn}(x) \) does not exist. Limits help us understand behavior near discontinuous points.

Graphing

Graphing the signum function can effectively display its step-like behavior, giving a visual understanding of its values. On a graph of \( \text{sgn}(x) \), you'll notice horizontal lines and a specific point:

• A horizontal line at \( y = -1 \) for all negative \( x \) values.
• A single point at \( (0, 0) \) to indicate that \( y = 0 \) precisely at \( x = 0 \).
• A horizontal line at \( y = 1 \) for all positive \( x \) values.

This graphical representation demonstrates the discontinuity at \( x = 0 \), as shown by the abrupt jump from -1 to 1.

Mathematical functions

The signum function is a prominent mathematical function characterized by its simplicity and utility in determining the sign of numeric input. It simplifies numerical analysis by converting values to

• -1 for any negative number,
• 0 when the number is precisely zero,
• and 1 for any positive number.

Despite appearing basic, it is a significant component in many mathematical and engineering applications. Its primary role begins in signal processing or when distinguishing between different regional values such as negatives, zeros, and positives. The step-like nature of the signum function makes it straightforward but with complex behaviors at transition points.

Continuity and Discontinuity

Continuity in a function means you can draw its graph without lifting your pen. However, the signum function is not continuous at \( x = 0 \). This is due to its distinct jumps, typical of a step function. Discontinuity happens here because:

• At \( x = 0 \), there is no gradual change; the function jumps from -1 to 1 directly.
• Since the left-hand limit at zero is -1 and the right-hand limit is 1, they are unequal, preventing the existence of \( \lim_{x \to 0} \text{sgn}(x) \).

Recognizing these discontinuities is crucial in calculus and when determining where a function does not smoothly connect. Understanding these points helps better grasp a function’s behavior and potential limitations.