Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. $$ \lim _{x \rightarrow 0^{+}} \frac{|x|}{x} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understanding the Problem

We are asked to find the limit of the function \( f(x) = \frac{|x|}{x} \) as \( x \) approaches 0 from the positive side, denoted as \( x \to 0^+ \). This involves analyzing the behavior of \( f(x) \) for values of \( x \) that are positive and close to zero.

Step 2: Analyzing the Function for Positive Values

When \( x > 0 \), it follows that \( |x| = x \). Therefore, for values of \( x \) approaching zero from the positive side, \( f(x) = \frac{x}{x} = 1 \). The function is constant in this region as \( x \) remains positive.

Step 3: Calculating the Right-Hand Limit

Because \( f(x) = 1 \) for all \( x > 0 \), as \( x \to 0^+ \), the function \( f(x) \) will also approach 1. Thus, the right-hand limit is found by evaluating \( \lim_{x \to 0^+} \frac{|x|}{x} = 1 \).

Key Concepts

Right-hand limit

A right-hand limit describes the behavior of a function as the input approaches a specific value from the right, or from values greater than the point of interest. In the context of our exercise, we're interested in the right-hand limit of the function \( f(x) = \frac{|x|}{x} \) as \( x \) approaches zero. This is denoted as \( x \to 0^+ \), meaning \( x \) is nearing zero while remaining positive.

• This kind of limit is focused on analyzing what happens just to the right of the point in question. For the function mentioned, it only considers small positive values very close to zero.
• Remember that if the function settles into a specific value as you approach this point, that value is the right-hand limit.

In this scenario, as we substitute smaller positive values into \( f(x) \), the function effectively evaluates consistently at 1. Therefore, the right-hand limit of \( \lim_{x \to 0^+} \frac{|x|}{x} \) is determined to be 1.

Absolute value function

The absolute value function is an interesting concept in mathematics that represents the distance of a number from the zero point on a number line, regardless of direction. This means:

• For a positive number \( x \), the absolute value is simply \( x \).
• For a negative number \( x \), it is the opposite of \( x \) (i.e., \( -x \)) to make sure the result is always positive.
• For zero, the absolute value is 0, since it's already at zero.

Using this understanding, when we use \(|x|\) as part of a function like \( \frac{|x|}{x} \), we observe how the absolute value influences the behavior of the function based on whether \( x \) is positive, negative, or zero. In the scenario provided, since \( x \) is approaching from the positive side, \( |x| = x \), simplifying the expression to \( \frac{x}{x} = 1 \). Notice how the absolute value operation ensures our function behaves consistently for positive values as \( x \to 0^+ \).

Approaching zero

"Approaching zero" is a fundamental concept when discussing limits, describing the action of a variable as it gets infinitely close to zero without actually reaching it. This idea is crucial for defining limits in calculus:

• It helps us understand the behavior of functions near points that might otherwise have undefined values.
• In the exercise, \( x \to 0^+ \) indicates \( x \) is coming close to zero from the right-hand side, or positive territory.
• This exploration doesn't require touching zero but rather examines the pattern as \( x \) becomes arbitrarily near it.

By analyzing the function \( \frac{|x|}{x} \) under these conditions, we infer the values that the function approaches, rather than what it might actually hold at zero. In practical terms, this technique helps solve potential points of discontinuity or ambiguity by looking at the limiting behavior.