Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit. Refer to the accompanying graph of \(y=f(x)\) when it is given. Do not use a calculator. $$\lim _{x \rightarrow 0^{+}} \frac{|x|}{x}$$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understanding the Function

The function given is \( f(x) = \frac{|x|}{x} \). The function \(|x|\) is the absolute value of \(x\), which is always non-negative. The value of \(\frac{|x|}{x}\) depends on whether \(x\) is positive or negative.

Step 2: Considering Right-Side Limit

Since we need the limit as \( x \to 0^+ \), we consider \(x\) approaching 0 from the right (i.e., from positive values). For \(x > 0\), \(|x| = x\), so \(\frac{|x|}{x} = \frac{x}{x} = 1\).

Step 3: Determining the Limit

As \(x\) approaches zero from the right ( x \to 0^+), the expression \( \frac{|x|}{x} \) consistently evaluates to 1 because \( x \) remains positive. Therefore, the limit is \( \lim_{x \to 0^+} \frac{|x|}{x} = 1 \).

Key Concepts

Absolute Value Functions

Absolute value functions are an essential mathematical concept. The absolute value of a number, denoted as \(|x|\), represents its distance from zero on the number line. This means the value is always non-negative. For instance, the absolute values of \(-3\) and \(+3\) are both \(|3|\), because both are just 3 units away from zero.
In the function \(|x|\), whenever \(x\) is a positive number or zero, \(|x|\) is simply \(x\) itself. However, if \(x\) is negative, \(|x|\) translates it to a positive number, essentially turning \(-x\) into \(x\). This property is what allows the function and concepts related to absolute values, like \( \frac{|x|}{x}\), to perform interesting tasks in mathematical analysis.
Returning to the function in our example, \( \frac{|x|}{x} \), this form yields values based on the sign of \(x\).

• If \(x > 0\), \( \frac{|x|}{x} = 1 \) because \(|x|\) equals \(x\).
• If \(x < 0\), \( \frac{|x|}{x} = -1 \) since \(|x|\) turns \(-x\) into \(x\), altering the fraction's value.

Understanding absolute value functions is crucial to navigating the nuances of mathematical problems, including those involving limits.

Right-Sided Limits

Right-sided limits are an important tool in calculus for analyzing the behavior of functions as they approach a particular point from the right side of the number line. We often express this concept using the notation \( \lim_{x \to a^+} f(x)\), signifying that \(x\) is approaching \(a\) from values greater than \(a\).
In our example problem with the limit \( \lim_{x \to 0^+} \frac{|x|}{x} \), we are interested in how the function behaves as \(x\) gets very close to zero, specifically from positive values. This is why we focus on the behavior of the function when \(x\) is slightly larger than zero.
Understanding this directionality:

• The notation \(0^+\) dictates that we only consider positive \(x\) values approaching zero, excluding zero from the computation.
• Since \(x > 0\), the absolute value \(|x|\) equals \(x\), making \( \frac{|x|}{x} = 1\).

As a result, continuously approaching zero from the right enables us to calculate that the right-side limit is 1. This analysis is crucial because it allows for precise interpretations of a function's behavior in specific contexts.

Graphical Analysis of Functions

Graphical analysis provides an intuitive approach to understanding functions and their limits. A graph can reveal complex aspects of behavior in a function over its domain. This includes identifying where the function approaches certain values and how it behaves as it reaches boundary points or changes in direction.
Considering the function \( f(x) = \frac{|x|}{x} \), the graphical representation can be particularly telling. For \(x > 0\), the function is a constant with value 1. For negative values of \(x\) (which we check outside of the problem but is good to know), it becomes \(-1\). This results in a graph with a distinct step-like appearance, featuring two horizontal lines:

• A horizontal line at \(y = 1\) for \(x > 0\).
• A horizontal line at \(y = -1\) for \(x < 0\).

Understanding this graph helps visualize the function's piecewise nature. When approaching zero from the right, the graph at \(x = 0\) distinctly approaches and remains at 1. This graphical representation can aid in comprehending right-side limits and offers a visual verification of the arithmetic results gained from algebraic computations.