Problem 43 - Calculus: Early Transcendentals

Question

Find the limit, if it exists. If the limit does not exist, explain why. \( \displaystyle \lim_{x \to 0.5^-}\frac{2x - 1}{| 2x^3 - x^2 |} \)

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Analyze the Limit Expression

We need to find \( \lim_{x \to 0.5^-}\frac{2x - 1}{| 2x^3 - x^2 |} \). Here, \( x \to 0.5^- \) means \( x \) approaches 0.5 from the left. The expression is a fraction with a numerator \(2x - 1\) and an absolute value in the denominator \( | 2x^3 - x^2 | \).

Step 2: Simplify Numerator and Examine the Denominator

As \( x \) approaches 0.5 from the left, observe that \( 2x - 1 \to 0 \), since \(2(0.5) - 1 = 0\). For \(2x^3 - x^2\), calculate at \( x = 0.5 \): \((2(0.5)^3 - (0.5)^2) = 2(0.125) - 0.25 = 0.25 - 0.25 = 0 \). The denominator approaches zero from negative values due to the cube term.

Step 3: Determine Sign of Denominator

Since \(2x^3 - x^2 = x^2(2x - 1)\) and as \(x\) approaches 0.5 from the left, \(2x - 1\) is slightly negative, making \(2x^3 - x^2\) negative right before it crosses zero from the left. Thus \(|2x^3 - x^2|\) becomes the absolute value, making the denominator positive right before 0.5.

Step 4: Apply Limit Laws and Evaluate

As \( x \to 0.5^-\), \(2x - 1\to 0^-\) and \( |2x^3 - x^2| \to 0^+\). The limit simplifies to \( \lim_{x \to 0.5^-} \frac{0^-}{0^+} = -\infty \) because a small negative number divided by a small positive number results in negative infinity when approaching zero.

Key Concepts

One-Sided Limits

Understanding one-sided limits is essential in evaluating expressions where a variable approaches a specific point from one direction. In this case, we deal with the limit as \( x \) approaches 0.5 from the left, denoted by \( x \to 0.5^- \). This means that \( x \) takes values slightly less than 0.5.

• One-sided limits allow us to see the behavior of a function as it approaches a point from only one direction.
• This is useful in analyzing functions that might behave differently when approaching a point from different directions.

For the expression \( \lim_{x \to 0.5^-}\frac{2x - 1}{| 2x^3 - x^2 |} \), we focus on how the numerator and denominator behave specifically as \( x \) gets closer to 0.5 from the left.

Absolute Value

Absolute value is crucial when dealing with expressions that result in negative values. It converts any negative result into its positive counterpart, by essentially measuring the 'distance' from zero.

• The absolute value function, denoted by \( |.| \), changes negative outputs to positive.
• In the expression \( |2x^3 - x^2| \), the denominator might initially take on negative values as \( x \) approaches 0.5 from the left.

However, with absolute value applied, \( |2x^3 - x^2| \) remains positive, which is crucial for determining the sign of the whole expression.

Infinite Limits

Infinite limits occur when expressions grow indefinitely or drop to very large negative values. An expression might approach infinity or negative infinity when a variable gets closer to a certain point but never actually reaches it.

• When assessing the expression \( \lim_{x \to 0.5^-} \frac{2x - 1}{| 2x^3 - x^2 |} \), note that the numerator \( 2x - 1 \) approaches 0 from the negative side.
• Meanwhile, the denominator approaches 0 from the positive side due to the absolute value.

This setup results in \(-\infty\) since a small negative number divided by a very tiny positive number makes the overall expression tend towards negative infinity.

Limit Laws

Limit laws are rules that guide us in evaluating complex limits by breaking them down into more manageable parts. These rules make calculations involving limits easier and more structured.

• The limit of a fraction is found by taking the limit of both the numerator and denominator separately.
• Applying the laws helps determine that as \( x \to 0.5^- \), the fraction approaches \(-\infty\) due to the signs of the numerator and denominator.

By evaluating both parts of the fraction and understanding their behavior near the limit point, we can confidently conclude that the limit does not exist in a finite form, but instead heads towards negative infinity.