Problem 33 - Precalculus Mathematics for Calculus

Question

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 0^{-}}\left(\frac{1}{x}-\frac{1}{|x|}\right)$$

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Understand the problem statement

We are asked to find the limit \( \lim _{x \rightarrow 0^{-}}\left(\frac{1}{x}-\frac{1}{|x|}\right) \). This means we need to evaluate the expression as \( x \) approaches 0 from the left (negative side).

Step 2: Simplify the expression

Since \( x \) is approaching 0 from the left, \( x \) is negative. Therefore, \(|x| = -x \). So the expression \( \frac{1}{x} - \frac{1}{|x|} = \frac{1}{x} - \frac{1}{-x} = \frac{1}{x} + \frac{1}{x} = \frac{2}{x}\).

Step 3: Evaluate the limit

Now that the expression is \( \frac{2}{x} \), consider what happens as \( x \) approaches 0 from the left. Since \( x \to 0^{-} \), \( x \) is a very small negative number, making \( \frac{2}{x} \) a very large negative number. Thus, the limit is \(-\infty\).

Step 4: Conclusion about the limit

Since \( \frac{2}{x} \) becomes unbounded as \( x \to 0^{-} \), the limit of the expression as \( x \) approaches 0 from the left is \(-\infty\).

Key Concepts

Left-Hand Limit

When dealing with limits in calculus, it's essential to distinguish which direction the variable approaches a certain value. In the given exercise, the expression is evaluated as \( x \) approaches zero from the left. We call this a left-hand limit.

The left-hand limit notation \( x \rightarrow 0^{-} \) specifies that we are interested in values of \( x \) that are less than zero but very close to zero. This nuanced approach tells us we're examining the limit from the negative side of zero.

Grasping the concept of left-hand limits is fundamental, as it allows us to explore the behavior of functions that might act differently when approached from various directions on the number line, potentially indicating discontinuities or unique characteristics.

Negative Side Approach

Understanding the directionality in limits is crucial in calculus. Approaching from the negative side implies considering values that are less than the target but approach it very closely.

The exercise asks us to look at \( x \rightarrow 0^{-} \), meaning \( x \) values are approaching zero from the negative side. For instance, numbers like \( -0.1, -0.01, \) or \( -0.001 \) are progressively closer to zero but always from the left.

Why does this matter? The negative side approach can lead to different limit evaluations compared to approaching from the positive side, especially in expressions involving variables in denominators or absolute values. It prepares us to handle more complex scenarios where such differences are pivotal.

Unbounded Behavior

Sometimes, as we approach a limit, the function doesn't settle at any finite value, instead moving toward infinity or negative infinity. This is referred to as unbounded behavior.

In the original exercise, simplifying the expression results in \( \frac{2}{x} \). As \( x \) approaches 0 from the negative side, this term grows excessively negative, leading to unbounded behavior.

• When \( x \) is a small negative, \( \frac{2}{x} \) becomes large but negative.
• This means that the function does not converge to a specific finite value.

In the context of limits, we express this by stating the limit is \(-\infty\). Recognizing unbounded behavior helps in identifying vertical asymptotes and understanding deeper aspects of function behavior near certain values.

Absolute Value Properties

The absolute value function presents a unique challenge in calculus, especially in limits, due to its piecewise nature. It treats positive and negative approaches differently.

For \( x < 0 \), the property \( |x| = -x \) becomes critical. This means when \( x \) approaches zero from the left, \( |x| \) needs to be considered as \( -x \).
This insight simplifies the exercise's expression \( \frac{1}{x} - \frac{1}{|x|} \) to \( \frac{2}{x} \).

• Without considering absolute value properties, simplifying such expressions would be challenging.
• Knowing how to handle absolute values ensures accurate evaluations.

In summary, understanding and applying absolute value properties are vital for precise limit calculations and deeper function analysis.