Problem 39 - 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 Limit Expression

We need to find the limit of the expression \( \frac{|x|}{x} \) as \( x \) approaches 0 from the left (denoted as \( 0^- \)). This means we're considering values of \( x \) that are slightly less than 0.

Step 2: Analyzing the Absolute Value

The expression \( |x| \) represents the absolute value of \( x \). When \( x \) is negative, the absolute value function changes the sign of \( x \) to positive, so \(|x| = -x\).

Step 3: Substituting Absolute Value in the Expression

Since \( x \) is approaching 0 from the negative side, substitute \(|x| = -x\) into the expression: \[ \frac{|x|}{x} = \frac{-x}{x} = -1 \]

Step 4: Evaluating the Limit

As \( x \) approaches 0 from the left, the expression \( \frac{-x}{x} \) simplifies to -1. This means: \[ \lim_{x \rightarrow 0^{-}} \frac{|x|}{x} = -1 \]

Key Concepts

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line without considering direction. In simple terms, for any real number \( x \):

• If \( x \) is positive or zero, \(|x| = x\).
• If \( x \) is negative, \(|x| = -x\), which converts the negative value into positive.

So why is absolute value important when dealing with limits? The absolute value function can change the behavior of the expression as \( x \) approaches certain points, such as zero. Understanding its properties helps us properly evaluate the expressions that include it.

In this exercise, recognizing that \( |x| = -x \) for \( x < 0 \) is crucial when finding the limit as \( x \) approaches 0 from the negative side.

Approaching From the Left

When we evaluate limits like \( \lim_{x \rightarrow 0^-} \), we focus on values of \( x \) that are slightly less than zero. This is called approaching from the left.

Visualize the number line. Approaching 0 from the left means moving from negative values toward zero. For example, from -0.1 to -0.01, and so on. This specific direction influences how certain functions behave near the point of interest.

In our exercise, as \( x \) approaches 0 from the left, \( x \) takes on negative values. Consequently, for the expression \( \frac{|x|}{x} \), the numerator becomes \(-x\) while the denominator is \(x\). Understanding this directional approach is key to correctly evaluating the limit.

Evaluating Limits Involving Absolute Values

Limits can seem challenging, especially with absolute values involved. The trick is to carefully analyze the behavior of each component as \( x \) approaches a specific value.

• First, assess the role of the absolute value for \( x < 0 \), where \( |x| = -x \).
• Next, substitute this into the expression, which transforms \( \frac{|x|}{x} \) into \( \frac{-x}{x} \).

The expression \( \frac{-x}{x} \) simplifies to -1, even as \( x \) gets infinitesimally close to 0. Hence, the overall limit is -1 as \( x \) approaches 0 from the left.

Remember, limits describe the behavior of a function as \( x \) nears a specified point. Breaking down the function using these steps helps us accurately find \( \lim_{x \rightarrow 0^-} \frac{|x|}{x} = -1 \).