Problem 41 - Calculus Early Transcendentals

Question

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 0.5^{-}} \frac{2 x-1}{\left|2 x^{3}-x^{2}\right|}$$

Step-by-Step Solution

Answer

The limit does not exist; it approaches negative infinity.

Step 1: Understand the Limit Expression

We are asked to find the left-hand limit as \( x \) approaches 0.5 for the function \( \frac{2x - 1}{|2x^3 - x^2|} \). This means we are looking at the behavior of the function when \( x \) is slightly less than 0.5.

Step 2: Evaluate the Numerator

The numerator of the expression is \( 2x - 1 \). As \( x \to 0.5^- \), the numerator becomes \( 2(0.5) - 1 = 1 - 1 = 0 \). So, the numerator approaches 0 as \( x \to 0.5^- \).

Step 3: Evaluate the Denominator

The denominator is \(|2x^3 - x^2|\). As \( x \to 0.5^- \), we compute \( 2(0.5)^3 - (0.5)^2 = 2 \times 0.125 - 0.25 = 0.25 - 0.25 = 0 \). Thus, \(|2x^3 - x^2|\) approaches 0.

Step 4: Examine the Behavior of the Denominator

Because the expression inside the absolute value approaches zero, check the sign for values just less than 0.5. If \( x < 0.5 \), \( 2x^3 - x^2 < 0.25 \) minus a larger negative value, which results in \( 2x^3 - x^2 < 0 \). Hence, \(|2x^3 - x^2| = -(2x^3 - x^2)\), causing the denominator to approach 0 from the negative side.

Step 5: Determine the Limit from Left-Hand Side

For \( x \to 0.5^- \), both the numerator and denominator approach 0. However, due to the absolute value property and signs as discussed, the overall fraction \( \frac{0}{-|0|} = 0/0 \) becomes undefined and approaches negative infinity.

Key Concepts

Left-Hand Limit

When calculating the left-hand limit, we focus on how a function behaves as the input approaches a specific value from the left-hand side. This is represented as \( x \rightarrow c^{-} \), indicating that \( x \) is getting closer to \( c \) from values less than \( c \).

In the context of the given problem, we explore what happens as \( x \) approaches 0.5 from the left for the function \( \frac{2x - 1}{|2x^3 - x^2|} \). As we analyze this, the terms in the function determine the resulting limit.

Understanding left-hand limits helps us understand local behavior in functions where small changes in \( x \), near a particular point, can significantly change the outcome. This is especially crucial in calculus when predicting the continuity and differentiability of a function.

Absolute Value Function

The absolute value function, represented by \( |x| \), is crucial when evaluating limits like in this problem. The absolute value returns the non-negative value of any real number, i.e., \( |x| = x \) if \( x \geq 0 \) and \( |x| = -x \) if \( x < 0 \).

In the original exercise, the function is \( |2x^3 - x^2| \). As we approach \( x = 0.5 \) from the left, we calculate that \( 2x^3 - x^2 \) becomes negative. Therefore, the absolute value function transforms the expression, leading us to consider \( |2x^3 - x^2| = -(2x^3 - x^2) \).

This transformation affects the denominator's direction and value, impacting the overall behavior of the limit. Understanding how absolute values adjust the behavior of functions is essential in analyzing limit problems.

Function Behavior

Evaluating function behavior as it approaches a limit involves understanding how both the numerator and the denominator change as \( x \) gets closer to a certain value. This involves analyzing both their magnitudes and signs.

For \( f(x) = \frac{2x - 1}{|2x^3 - x^2|} \) as \( x \rightarrow 0.5^{-} \), both the numerator \( 2x - 1 \) and the denominator \( |2x^3 - x^2| \) approach zero. However, these elements changing simultaneously to zero creates an indeterminate form \( \frac{0}{0} \), leading us to examine the changes more closely.

The signs and absolute value transformation suggest that the denominator approaches zero from the negative side, causing the fraction to potentially reach negative infinity. Such insights are valuable for predicting limits in complex functions and understanding the nuanced behaviors they may exhibit.

Undefined Limit

An undefined limit arises when we cannot determine a specific value that a function approaches. This often occurs with indeterminate forms like \( \frac{0}{0} \), where both the numerator and denominator approach zero.

In this exercise, the left-hand limit of \( \lim_{x \rightarrow 0.5^{-}} \frac{2x - 1}{|2x^3 - x^2|} \) is undefined since both components go to zero. More specifically, the behavior of the components causes the function to attempt reaching negative infinity rather than approaching a finite value.

When the limit is undefined like in this case, it typically signals that the graph of the function has a vertical asymptote or that the function behaves irregularly near the point of interest. Such insights help predict how and why functions behave fundamentally differently near certain critical values.