Problem 14 - 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.

Step 1: Analyze the Function

The function given is \( \frac{|x|}{x} \). When \( x < 0 \), the absolute value \( |x| \) becomes \( -x \). So, the function simplifies to \( \frac{-x}{x} = -1 \).

Step 2: Determine Behavior as x Approaches 0 from the Left

Since we are considering \( x \to 0^{-} \), meaning \( x \) approaches 0 from the left (negative side), every \( x \) is negative in this case. Thus, the expression \( \frac{|x|}{x} \) simplifies to \(-1\) for values of \( x \) approaching 0 from the negative side.

Step 3: Calculate the Limit

Since the simplified function value is constantly \(-1\) for every negative \( x \) and does not depend on the specific value of \( x \) as it approaches 0, the limit is constant and equal to \(-1\). Therefore, the limit of \( \lim_{x \rightarrow 0^{-}} \frac{|x|}{x} \) is \(-1\).

Key Concepts

Absolute Value

The concept of absolute value is central to our problem, as it affects how functions behave, especially when they include both positive and negative numbers. The absolute value of a number is its distance from zero on the number line, regardless of direction.
This means for any real number, expressed as \(x\), its absolute value is denoted by \(|x|\).

• For positive numbers, \(|x| = x\).
• For negative numbers, \(|x| = -x\).

This even-handed treatment of positive and negative numbers is what makes absolute value such a versatile tool in limit problems. In our function \(\frac{|x|}{x}\), you can see how the absolute value transforms \(|x|\) to \(-x\) when \(x < 0\). This step simplifies the function, allowing us to find the limit easily as \(x\) approaches 0 from the left.

Left-Hand Limit

A left-hand limit examines the behavior of a function as the input approaches a specific point from the left. This is denoted by the symbol \(-\) right after the limit point, for example \( \lim_{x \to 0^{-}} \).
This is a crucial concept for understanding piecewise functions and functions that may behave differently depending on the direction of approach.

• The term \(x \to 0^{-}\) specifies approaching zero from values less than zero.
• If a function results in a constant value when approached from the left, the left-hand limit equals that constant.

In the given example, as \(x\) approaches zero from the negative side, \(|x|\) converts to \(-x\). For all negative \(x\) values near zero, the function \(\frac{|x|}{x}\) becomes \(-1\), demonstrating that the left-hand limit is \(-1\).

Epsilon-Delta Method

The epsilon-delta method is a formal procedure for proving the limit of a function, often used in proving that a limit exists and is of a certain value. However, in some straightforward cases, an epsilon-delta test may not be necessary. Our exercise is one of these cases where the limit is clear from simplification and logical reasoning.
Here's a simplified overview of the epsilon-delta method for context:

• \(\epsilon\) represents how close we want the function value \(f(x)\) to be within some limit \(L\).
• \(\delta\) represents the distance \(x\) is from \(a\), such that whenever \(0 < |x - a| < \delta\), we ensure \(|f(x) - L| < \epsilon\).

In this problem, we recognize that as \(x\) approaches zero from the left, the value of the function is consistently \(-1\). As such, a function like this that simplifies cleanly doesn't require epsilon-delta validation to recognize the limit is \(-1\). This understanding saves time and effort when encountering similar mathematical problems. It's essential to know when a problem can be tackled head-on, leveraging simpler logic over formal proofs.