Problem 44 - Calculus Early Transcendentals

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 0.

Step 1: Understand the Function

We are given the function \( f(x) = \frac{1}{x} - \frac{1}{|x|} \). This function seems problematic only when \( x \) approaches zero because of the expressions \( \frac{1}{x} \) and \( \frac{1}{|x|} \) involving division by zero.

Step 2: Analyze the Absolute Value

Since we are taking the right-hand limit (\( x \rightarrow 0^{+} \)), the absolute value in \( |x| \) can be ignored because \( |x| = x \) as \( x \) is positive when approaching zero from the right.

Step 3: Simplify the Function

Substituting \( |x| = x \) for \( x \rightarrow 0^{+} \), the expression becomes \( \frac{1}{x} - \frac{1}{x} = 0 \). This simplifies the function \( f(x) \) to 0.

Step 4: Evaluate the Limit

Since \( f(x) = 0 \) for all \( x > 0 \), the limit \( \lim_{x \rightarrow 0^{+}} f(x) = \lim_{x \rightarrow 0^{+}} 0 = 0 \). Thus, the limit exists and is equal to zero.

Key Concepts

One-sided Limit

When we talk about limits, we often consider the behavior of a function as the input approaches a particular point. A one-sided limit is a specific type of limit where we only look from one direction—either from the left or the right. In this case, we are evaluating the right-handed or right-sided limit, expressed as \( \lim_{x \rightarrow 0^{+}} \).

This indicates that we're interested in the behavior of the function as the variable \( x \) approaches 0 from values greater than 0. It's an important concept because sometimes the behavior as we approach from the right is different from approaching from the left. This gives us a more nuanced understanding of a function's behavior at a point.

• The right-sided limit is expressed with a plus sign in the exponent, like \( x \rightarrow 0^{+} \).
• Understanding one-sided limits is crucial for functions that behave differently on each side of a point.
• In this exercise, we're not concerned about negative values, as it's a right-hand limit.

Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering direction. This means the absolute value is always positive, regardless of whether the original number was negative or positive.

For the function in the exercise, the absolute value signifies something significant. The expression \( |x| \) simply turns \( x \) into a positive value. As \( x \) approaches zero from the positive side (i.e., positive values closer and closer to zero), \( |x| = x \).

• Absolute value is represented with vertical bars, like \( |x| \).
• On the positive side, \( |x| = x \) when \( x \) is positive.
• In our limit, this means we treat \( |x| \) exactly like \( x \) when approaching zero from the positive side.

Approaching Zero

In limits, approaching a value means getting closer and closer to it without necessarily ever reaching it. When we say \( x \rightarrow 0^{+} \), we are considering \( x \) values that are increasingly smaller and positive, inching their way toward zero.

This concept is pivotal in both calculus and analysis because it lets us examine the behavior of functions near specific points without needing those functions to actually "arrive" at the points.

• Approaching zero doesn't mean reaching zero; it means getting very close.
• In one-sided limits, approaching can happen from above (positive) or below (negative).
• It helps assess what happens to a function or an equation around critical points.

Limit Evaluation

Evaluating a limit takes us through evaluating the behavior of a function as the variable gets close to a certain point. In our problem, we examine \( \lim_{x \rightarrow 0^{+}} \left( \frac{1}{x} - \frac{1}{|x|} \right) \).

By simplifying the expression with \( |x| = x \) (since we're only looking at positive \( x \)), the function becomes \( \frac{1}{x} - \frac{1}{x} \), which simplifies further to 0. This shows that as \( x \) gets closer to zero from the right, the difference between these fractions becomes zero.

• Limit evaluation involves simplification and careful analysis of expressions.
• Recognize when certain expressions cancel each other out, as they do here with \( \frac{1}{x} - \frac{1}{x} = 0 \).
• Show the final answer clearly: the limit is zero in this context.