Problem 13 - Calculus

Question

Find the limits if they exist. An \(\varepsilon-\delta\) test is not required. $$ \lim _{x \rightarrow 0^{+}} \frac{|x|}{x} $$

Step-by-Step Solution

Answer

The limit is 1 as \( x \to 0^+ \).

Step 1: Understanding the Expression

The expression \( |x|/x \) represents the absolute value of \( x \) divided by \( x \). Since \( |x| \) equals \( x \) when \( x \) is positive, this will simplify our expression.

Step 2: Substitute Values Approaching from the Right

As \( x \) approaches 0 from the right (i.e., \( x \rightarrow 0^+ \)), \( x \) is positive. So, \( |x| = x \) and the expression \( |x|/x = x/x = 1 \).

Step 3: Evaluate the Limit for Positive Approach

When approaching 0 from the right, since the expression simplifies to 1, the limit is \( \lim_{x \rightarrow 0^{+}} \frac{|x|}{x} = 1 \).

Key Concepts

Epsilon-Delta Definition

In calculus, limits are a fundamental concept, and the epsilon-delta definition provides a precise way to describe them. This definition is used to show that a function approaches a particular value as the input gets very close to some point. The idea is to demonstrate that for every small number (\( \varepsilon \)) you choose, there exists another small number (\( \delta \)) such that if the distance between \( x \) and the point of interest is within \( \delta \), then the distance between the function value and the limit is within \( \varepsilon \).
This rigorous approach helps eliminate any ambiguity about the behavior of the function near the given point.
However, when solving problems by inspection, especially simple ones, this detailed approach isn’t always needed. In our provided exercise, you don’t need the epsilon-delta test, as analyzing the function behavior on either side of the limit is enough.
But knowing this definition is crucial, as it builds the foundation for more advanced mathematical reasoning.

One-Sided Limits

One-sided limits consider the behavior of a function as it approaches a particular value only from one direction, either from the left or the right. In our problem, the one-sided limit is from the right, noted by the \( x \rightarrow 0^{+} \). This notation indicates that \( x \) is approaching 0 from positive values, but not reaching or considering any negative values.
This is crucial because functions may behave differently from either side. For example, the absolute value function \( |x|/x \) demonstrates this clearly:

• If \( x \) approaches 0 from the positive side, \( |x| = x \), resulting in \( 1 \).
• If \( x \) were to approach 0 from the negative side, \( |x| = -x \), leading to \( -1 \).

Understanding one-sided limits helps in correctly evaluating the expression and determining the function's true behavior at discontinuities or points of interest.

Absolute Value Function

The absolute value function, denoted as \( |x| \), represents the distance of a number from zero on the number line, returning only non-negative values.
So \( |x| = x \) if \( x \geq 0 \) and \( |x| = -x \) if \( x < 0 \). This function plays a significant role in many areas of mathematics, including limits.
In the original exercise, \( |x|/x \), the absolute value impacts the outcome of the limit depending on the direction from which \( x \) approaches zero.

• For \( x \rightarrow 0^+ \), \( |x| = x \) leads to the expression being \( 1 \).
• Conversely, if it were \( x \rightarrow 0^- \), the function would yield \( -1 \).

Recognizing how the absolute value function adjusts based on input direction allows us to accurately determine limits, especially in situations involving discontinuities or abrupt changes in function behavior.