Problem 46 - Calculus: Early Transcendentals

Question

Find the limit, if it exists. If the limit does not exist, explain why. \( \displaystyle \lim_{x \to 0^+}\left(\frac{1}{x} - \frac{1}{|x|} \right) \)

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Analyze the Absolute Value Function

Consider the behavior of the absolute value function when approaching 0 from the positive direction. For any positive number \( x \), the absolute value function \( |x| \) is equal to \( x \). Therefore, \( \frac{1}{|x|} = \frac{1}{x} \) when \( x > 0 \).

Step 2: Simplify the Expression

Substitute the expression from Step 1 into the limit expression: \( \frac{1}{x} - \frac{1}{|x|} = \frac{1}{x} - \frac{1}{x} \). This simplifies to 0 for all \( x > 0 \).

Step 3: Evaluate the Limit

Evaluate the simplified expression: \( \lim_{x \to 0^+}(0) = 0 \). The limit of a constant is simply that constant itself.

Key Concepts

Absolute Value

The concept of Absolute Value is crucial when analyzing mathematical functions, especially when solving limits. Absolute value, represented as \(|x|\), refers to the non-negative value of \(x\). It measures the distance of a number from zero on the number line, ignoring whether the number is positive or negative. For instance, \(|3| = 3\) and \(|-3| = 3\).

In the context of limits and calculus, understanding the absolute value function helps in evaluating expressions as variables approach particular points. In this problem, since \(x\) is approaching 0 from the positive side, we have \(|x| = x\). This simplification allows us to reduce complex expressions, as seen in the exercise's solution, where \(|x|\) behaved identically to \(x\) for positive values.

Right-hand Limit

The Right-hand Limit is a type of limit that considers the behavior of a function as the variable approaches a specific value from the right side. It is denoted by a superscript plus, such as \(x \to 0^+\). This indicates that \(x\) is approaching 0 from values that are greater than 0, just ever-so-slighly to the right of zero on the number line.

Calculating a right-hand limit involves considering only those values of \(x\) that are on this right side, which can greatly simplify problem solving. In this example, since we are only concerned with \(x\) values greater than 0, we leverage the property of absolute value such that \(|x| = x\). This simplification directly impacted the limit evaluation, making it much easier to find the limit as \(x\to 0^+\).

Right-hand limits are important for understanding piecewise functions and situations where the function behaves differently on either side of a point.

Limit Evaluation

Limit Evaluation is a fundamental concept in calculus, which involves assessing the value that a function approaches as the input gets closer to some specified point. Limits are the backbone of defining derivatives and integrals. In evaluating limits, especially when there are absolute values or right-sided limits involved, simplification can be key.

In this exercise, once we simplified the expression \( \frac{1}{x} - \frac{1}{|x|} \) to 0, evaluating the limit became straightforward. It highlighted how, by understanding and simplifying the components of the function, the limit evaluation becomes much easier to perform. For instance, since \( \lim_{x \to 0^+}(0) = 0 \), we conclude that no matter how close \(x\) gets to 0 from the positive side, the result of the expression remains 0.

This concept stresses the importance of simplification and understanding the behavior of components within a function to accurately predict the limit. It's a fundamental toolset for further explorations in calculus.