Problem 35 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(31 - 36 ,\) determine the limit. $$\lim _ { x \rightarrow 0 ^ { + } } \frac { x } { | x | }$$

Step-by-Step Solution

Answer

The limit of the function \( \frac { x } { | x | } \) as \( x \) approaches 0 from the right is 1.

Step 1: Understand the absolute value function

For any real number \( x \), the absolute value \( |x| \) is defined as \( x \) if \( x \geq 0 \), and \(-x\) if \( x < 0 \). As \( x \) is approaching 0 from the right (positive direction), we have \( x > 0 \), thus \( |x| = x \).

Step 2: Substitute the absolute value function in the given function

Substitute \( |x| \) with \( x \). So, the given function \( \frac { x } { | x | }\) now becomes \( \frac { x } { x }\).

Step 3: Simplify the function

The function \( \frac { x } { x }\) simplifies to 1 for all \( x \neq 0 \). So, the given function is equal to 1 when \( x > 0 \).

Step 4: Find the limit

Now that the function has been simplified, find the limit as \( x \) approaches 0 from the right. Since the function is 1 for all \( x > 0 \), the limit is 1.

Key Concepts

Understanding the Absolute Value Function

The absolute value function, denoted as \( |x| \), is a critical mathematical concept used throughout various areas of mathematics, including calculus. It represents the distance of a number from zero on a number line without considering direction. In simpler terms, it turns negative numbers to positive ones, while positive numbers stay unchanged. This function is defined piecewise:

\( |x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases} \)

This means if you're working with a positive number, the absolute value function is just the number itself. For a negative number, it's the number multiplied by -1. Understanding this function is crucial when simplifying expressions and determining limits, particularly when the limit approaches a point from one side—a concept known as the unilateral limit.

Interpreting Unilateral Limits

A unilateral limit, also referred to as a one-sided limit, involves the behavior of a function as the variable approaches a particular value from one side, either from the left (\( x \rightarrow c^- \)) or from the right (\( x \rightarrow c^+ \)). In the context of our exercise:

\( \lim_ { x \rightarrow 0 ^ { + } } \frac { x } { | x | } \)

the notation \( 0^+ \) indicates we're interested in the behavior of the function as \( x \) gets infinitesimally close to zero from the positive side. This is significant because, while limits typically consider the approach from both sides, unilateral limits focus on just one direction, and this can impact the limit's value, especially when dealing with functions like the absolute value that behave differently on each side of a point.

Simplifying Expressions in Calculus

Simplifying expressions is a foundational skill in calculus. It involves rewriting expressions in a more manageable form, often to facilitate the application of calculus principles like taking derivatives or evaluating limits. In practical terms, it can mean canceling out terms, factoring, expanding polynomials, or dealing with complex fractions. For the given exercise:

\( \frac { x } { | x | } \)

we simplify by recognizing that, for \( x > 0 \), the expression \( |x| \) is equivalent to \( x \). Thus, the function simplifies to \( \frac { x } { x } \), which further simplifies to 1, because any non-zero number divided by itself equals 1. Note that simplification should always be done with an eye toward the domain of the function to avoid mistakes such as division by zero.