Problem 39 - Contemporary Precalculus

Question

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 0}\left[\frac{|x|}{x}-\frac{x}{|x|}\right]$$

Step-by-Step Solution

Answer

Answer: The limit of the function \(\left[\frac{|x|}{x}-\frac{x}{|x|}\right]\) as x approaches 0 is 0.

Step 1: Separate into Cases

We'll need to consider the function approaching 0 from the left and the right. Let's first break down each expression in the brackets separately. For \(\frac{|x|}{x}\): - If \(x > 0\), \(\frac{|x|}{x} = \frac{x}{x} = 1\). - If \(x < 0\), \(\frac{|x|}{x} = \frac{-x}{x} = -1\) . For \(\frac{x}{|x|}\): - If \(x > 0\), \(\frac{x}{|x|} = \frac{x}{x} = 1\). - If \(x < 0\), \(\frac{x}{|x|} = \frac{x}{-x} = -1\). Now we can analyze the given expression for each case.

Step 2: Analyze for \(x \rightarrow 0^+\)

As x approaches \(0\) from the right side, we have: $$\lim _{x \rightarrow 0^+}\left[\frac{|x|}{x}-\frac{x}{|x|}\right] =\lim _{x \rightarrow 0^+}\left[1 - 1\right] = 0$$

Step 3: Analyze for \(x \rightarrow 0^-\)

As x approaches \(0\) from the left side, we have: $$\lim _{x \rightarrow 0^-}\left[\frac{|x|}{x}-\frac{x}{|x|}\right] =\lim _{x \rightarrow 0^-}\left[-1 - (-1)\right] = 0$$

Step 4: Determine the Limit

Since both limits from left and right approached 0, the overall limit exists as well: $$\lim _{x \rightarrow 0}\left[\frac{|x|}{x}-\frac{x}{|x|}\right] = 0$$

Key Concepts

Absolute Value Functions

Absolute value functions are expressions that give us the magnitude of a number without considering its sign. The absolute value of a number \( x \) is written as \( |x| \). This function returns \( x \) if \( x \) is positive or zero, and \(-x\) if \( x \) is negative.
For example:

• If \( x = 3 \), then \( |x| = 3 \).
• If \( x = -3 \), then \( |x| = 3 \).

In the context of limits and calculus, absolute value functions can create piecewise definitions, as the behavior of the function changes at the point \( x = 0 \). This change in behavior makes analyzing limits near point \( x = 0 \) interesting, as we need to account for these different behaviors based on the direction from which \( x \) approaches zero.

Piecewise Functions

Piecewise functions are defined by multiple sub-functions, each of which applies to a particular interval of the main function's domain. This means that the output of the function depends on the specific region or interval you are evaluating.

In the case of the function categorized by absolute values, we consider two cases:

• When \( x > 0 \), absolute value expressions simplify differently than when \( x < 0 \).
• For example, \( \frac{|x|}{x} \) switches from 1 to -1 as \( x \) passes through zero.

When analyzing limits involving piecewise functions, it's crucial to evaluate the function's definition in each interval close to the point of interest. This will help you determine if the limit exists, non-existent or needs further explanations.

Limit Existence

The concept of limit existence is central in calculus, determining whether a specific limit will give a finite result as a variable approaches a point. For a limit to exist, the left-hand and right-hand limits at a given point must be equal.

Consider the expression:\[ \lim _{x \rightarrow 0}\left[\frac{|x|}{x}-\frac{x}{|x|}\right] \]The process involves checking values as \( x \) approaches 0 from both positive and negative directions. If both directions yield the same value (in this case, zero), we can confidently say that the limit exists. Otherwise, if the results differ, the limit does not exist at that point.

Left-Hand and Right-Hand Limits

In calculus, evaluating the left-hand and right-hand limits helps in understanding the behavior of functions near a particular point of interest. Specifically, these concepts help in establishing the existence of a limit.

To find the left-hand limit as \( x \rightarrow 0^- \), you evaluate the function as it approaches 0 from negative values. Conversely, to find the right-hand limit as \( x \rightarrow 0^+ \), you analyze the function's behavior as \( x \) approaches 0 from positive values.
The given expression:- **Left-Hand Limit**: \( \lim _{x \rightarrow 0^-}\left[\frac{|x|}{x}-\frac{x}{|x|}\right] = 0 \) - **Right-Hand Limit**: \( \lim _{x \rightarrow 0^+}\left[\frac{|x|}{x}-\frac{x}{|x|}\right] = 0 \)Both limits equating to zero indicates the function behaves consistently from both sides approaching the point \( x = 0 \), confirming that the overall limit exists.