Problem 44 - Precalculus

Question

For the following exercises, use a graphing utility to find numerical or graphical evidence to determine the left and righthand limits of the function given as \(x\) approaches \(a\). If the function has a limit as \(x\) approaches \(a\), state it. If not, discuss why there is no limit. $$\lim _{x \rightarrow 0} \frac{|x|}{x}$$

Step-by-Step Solution

Answer

The function does not have a limit at \( x = 0 \) because the left and right limits differ.

Step 1: Understanding the Expression

We need to analyze the expression \( \lim_{x \to 0} \frac{|x|}{x} \). The function is defined as \( f(x) = \frac{|x|}{x} \). This function behaves differently for positive and negative values of \( x \), as the absolute value function changes its behavior across zero.

Step 2: Evaluate Right-hand Limit

We evaluate the limit as \( x \to 0^+ \), which involves taking \( x \) values just greater than 0. Since \( x \) is positive, \( |x| = x \). Thus, \( \frac{|x|}{x} = \frac{x}{x} = 1 \). Therefore, \( \lim_{x \to 0^+} \frac{|x|}{x} = 1 \).

Step 3: Evaluate Left-hand Limit

Now we examine the limit as \( x \to 0^- \), taking \( x \) values just less than 0. Here, \( x \) is negative, so \( |x| = -x \). Therefore, \( \frac{|x|}{x} = \frac{-x}{x} = -1 \). Thus, \( \lim_{x \to 0^-} \frac{|x|}{x} = -1 \).

Step 4: Compare Left and Right Limits

The right-hand limit of \( \frac{|x|}{x} \) as \( x \to 0 \) is 1, while the left-hand limit is -1. Since these two results are not equal, the limit \( \lim_{x \to 0} \frac{|x|}{x} \) does not exist.

Key Concepts

Right-hand limit

When talking about the right-hand limit in calculus, we focus on what the value of a function approaches as the independent variable approaches a certain point from the right side. In simple terms, imagine approaching a destination from the east (right side) on a map. You are moving closer and closer to your destination without quite reaching it. Mathematically, this is represented as \ \( \lim_{{x \to a^+}} f(x) \ \). The notation \( a^+ \) means we're considering values of \( x \) that are just greater than \( a \). For our specific function \( f(x) = \frac{|x|}{x} \), when \( x \) approaches zero from the right, the absolute value helps determine its effect. Since \( x \) is positive just above zero, \( |x| = x \). This simplifies the function to \( \frac{x}{x} = 1 \). Thus, the right-hand limit of this function as \( x \to 0^+ \) is 1.

• Approaching Right: From values greater than target \( x \).
• Simplifying Absolute Value: Positive \( x \) means \( |x| = x \).
• Result: Gives the fraction \( \frac{x}{x} = 1 \).

Left-hand limit

The left-hand limit is similar but from the opposite side. It analyzes what happens to the function as the variable approaches a point from the left, or west side, of our imaginary map. We write this mathematically as \ \( \lim_{{x \to a^-}} f(x) \ \), where \( a^- \) indicates that we're using numbers slightly less than \( a \). Applying this to the function \( \frac{|x|}{x} \), as \( x \) nudges closer to zero from the negative side, \( x \) is negative. Therefore, \( |x| = -x \), which flips the function into \( \frac{-x}{x} = -1 \). So, the left-hand limit as \( x \to 0^- \) is -1.

• Approaching Left: From values less than target \( x \).
• Simplifying Absolute Value: Negative \( x \) means \( |x| = -x \).
• Result: Leads to \( \frac{-x}{x} = -1 \).

Absolute value function

The absolute value function is fundamental in redefining values within a range to always be positive. It represents the distance of a number on the number line from zero, regardless of direction. This is shown as \( |x| \), which equals \( x \) when \( x \) is positive or zero, and \(-x\) when \( x \) is negative. The behavior of absolute value is crucial in piecewise functions that have different values in different intervals of the domain.
This aspect is crucial for identifying limits, as it creates different behaviors for different directional approaches. When calculating limits such as \( \lim_{{x \to 0}} \frac{|x|}{x} \):

• For \( x > 0 \): \( |x| = x \), leading to positive results.
• For \( x < 0 \): \( |x| = -x \), leading to negative outcomes.
• Outcome: Helps determine if a limit exists or not when combined with \( \frac{|x|}{x} \).

This behavior explains why the absolute value function results in two different limit values, one from each direction.