Problem 45 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

(i) 1; (ii) -1; (iii) Does not exist; (iv) 1.

Step 1: Graph the Signum Function

The signum function, \( \operatorname{sgn} x \), is defined in piecewise segments. For \( x < 0 \), the value is \(-1\). For \( x = 0 \), the value is \(0\). For \( x > 0 \), the value is \(1\). To sketch this function, plot a horizontal line at \( y = -1 \) for all \( x < 0 \), a single point at the origin as \( y = 0 \), and another horizontal line at \( y = 1 \) for all \( x > 0 \).

Step 2: Find Limit as x Approaches 0 from the Right

To find \( \lim_{x \rightarrow 0^{+}} \operatorname{sgn} x \), consider the behavior of \( \operatorname{sgn} x \) as \( x \) approaches 0 from positive numbers. For \( x > 0 \), \( \operatorname{sgn} x \) is 1. Thus, the limit is 1.

Step 3: Find Limit as x Approaches 0 from the Left

To find \( \lim_{x \rightarrow 0^{-}} \operatorname{sgn} x \), consider \( \operatorname{sgn} x \) as \( x \) approaches 0 from negative numbers. For \( x < 0 \), \( \operatorname{sgn} x \) is -1. Thus, the limit is -1.

Step 4: Evaluate Two-Sided Limit at x = 0

For \( \lim_{x \rightarrow 0} \operatorname{sgn} x \), compare the limits from the left and right. Since \( \lim_{x \rightarrow 0^{+}} \operatorname{sgn} x = 1 \) and \( \lim_{x \rightarrow 0^{-}} \operatorname{sgn} x = -1 \), the two-sided limit does not exist, because the left and right limits are not equal.

Step 5: Find Limit at x = 0 for Absolute Value of Signum Function

For \( \lim_{x \rightarrow 0} |\operatorname{sgn} x| \), observe \(|\operatorname{sgn} x|\) is 1 for both \( x > 0 \) and \( x < 0 \), and 0 for \( x = 0 \). Therefore, both left and right limits and the value at zero are \( 1 \), making the limit \( 1 \).

Key Concepts

Limits

In calculus, limits help us understand the behavior of functions as input values approach a particular point. When dealing with the signum function, it's important to consider the behavior of the function as it approaches zero from both directions.

- **Right-hand limit**: As you approach zero from the positive side (\( x \rightarrow 0^{+} \)), the signum function keeps outputting 1. Thus, the limit is 1.
- **Left-hand limit**: As you approach zero from the negative side (\( x \rightarrow 0^{-} \)), the signum keeps buckling out -1. Here, the limit proves to be -1.

These two pieces of information tell us something very interesting: the two-sided limit does not exist because the values from either side differ. This is key when the left-hand and the right-hand limits don’t align.

Piecewise Functions

Piecewise functions are comprised of different sections, each with its own rule. Each interval operates independently based on specific conditions. The signum function is a classic example of a piecewise function.

- For any negative input, it returns -1.
- At exactly zero, the output is 0.
- If the input is positive, it results in 1.

The function acts differently under various circumstances. In each covered segment, the function exhibits unique characteristics based on directives in its definition, illustrating how different tools of math can piece together under one umbrella.

Graphing

Graphing the signum function helps visualize its characteristics and behavior. This is particularly useful for interpreting piecewise-defined functions. When sketching, follow simple steps:

- Draw a consistent horizontal line at \( y = -1 \) for all negative values of \( x \).
- For \( x = 0 \), plot a single point at the origin where \( y = 0 \).
- Draw another line at \( y = 1 \) for positive \( x \) values.

This graphical representation provides a clear breakdown of how the function responds to inputs, laying the groundwork for understanding limits, continuity, and other characteristics at a balance.

Discontinuity

Discontinuity occurs when a function makes abrupt changes at certain points. For the signum function, the discontinuity is centered around zero. Here’s why:

• When approaching zero from the negative side, the output is -1.
• From the positive side, it switches to 1.
• There’s a sudden leap in the function’s value directly at zero.

Since the values from either side do not converge, the function simply jumps from -1 to 1, with no smooth transition. This kind of behavior underscores a classic example of discontinuity, emphasizing where traditional calculus cannot define limits across sudden changes.