Problem 32 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understanding the Problem

We need to find the limit of the expression \( \frac{1}{x} - \frac{1}{|x|} \) as \( x \) approaches 0 from the right (i.e., \( x \rightarrow 0^{+} \)). The challenge is to understand the behavior of this expression near 0 from the positive side.

Step 2: Simplifying the Expression for Positive \( x \)

Since \( x \rightarrow 0^{+} \), \( x \gt 0 \). Therefore, \(|x| = x\). Substitute \(|x| = x\) in the expression: \( \frac{1}{x} - \frac{1}{x} = 0 \).

Step 3: Evaluating the Limit

The expression simplifies to \(0\) for any \( x > 0 \). Thus, as \( x \) approaches 0 from the right, the expression becomes constantly zero. Therefore, the limit is \( \lim_{x \rightarrow 0^{+}} 0 = 0 \).

Key Concepts

Limit Evaluation

When tackling problems involving limits, the goal is to determine the value that a function approaches as the input comes arbitrarily close to a certain point. In our exercise, we are asked to evaluate the limit of \( \frac{1}{x} - \frac{1}{|x|} \) as \( x \) approaches 0 from the positive side, notated as \( x \rightarrow 0^{+} \). This notation indicates we are only considering values of \( x \) that are slightly greater than zero. To evaluate limits, we can:

• Simplify the function to easily recognize patterns or behaviors as \( x \) reaches the designated point.
• Approach the problem using numerical, algebraic, or graphical methods to observe trends.
• Ensure that operations such as substitution do not lead to undefined results or indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

In our specific problem, simplifying the expression by considering the behavior of \( |x| \) for \( x > 0 \) leads to a straightforward solution, confirming that the limit is zero.

Right-Hand Limit

The notion of a right-hand limit focuses on approaching a particular value exclusively from the right. For example, with our problem, the expression \( \frac{1}{x} - \frac{1}{|x|} \) is evaluated as \( x \rightarrow 0^{+} \). This is necessary when the left and right behaviors of the function near a point may be different or undefined, and we are interested only in the behavior approaching from the right. Right-hand limits are expressed using the notation \( \lim_{x \rightarrow a^{+}} f(x) \), where \( a \) is the point of interest. It means we look only at values where \( x > a \). This is helpful in:

• Discontinuous functions where the behavior is different when approached from different sides.
• Piecewise functions that define different expressions on different intervals.

In our exercise, considering the right-hand limit ensures we only examine situations where \( x \) is a positive number approaching zero, leading to a clearer, more accurate evaluation of the expression.

Absolute Value Function

The absolute value function \( |x| \) returns the non-negative value of \( x \). It effectively removes any negatives by turning any negative input into its positive counterpart. In the expression \( \frac{1}{x} - \frac{1}{|x|} \), understanding the role of \( |x| \) is crucial for evaluating the limit as \( x \) approaches 0 from the right.For \( x > 0 \), the absolute value of \( x \) is simply \( x \) itself, meaning \( |x| = x \). This knowledge simplifies our expression to \( \frac{1}{x} - \frac{1}{x} = 0 \). Key takeaways about absolute value functions include:

• They affect the sign of the results and make certain limits easier to evaluate, as they can simplify complex expressions.
• In piecewise functions, where definitions change based on the sign of \( x \), the absolute value can ease the transition across functions.

In our limit evaluation, recognizing that \( |x| \) and \( x \) are equal for positive values allows us to simplify and solve the limit accurately, showcasing the importance of understanding absolute value behavior.