Problem 18 - Calculus

Question

Does \(f(x)=\frac{|x|}{x}\) have right or left limits at \(0 ?\) Is \(f(x)\) continuous?

Step-by-Step Solution

Answer

\(f(x)\) has different right and left limits at 0, so it's discontinuous there.

Step 1: Understand the Function

The function given is \(f(x) = \frac{|x|}{x}\). This function is undefined at \(x = 0\) because it results in a division by zero. For positive \(x\), \(|x| = x\) and thus \(f(x) = 1\). For negative \(x\), \(|x| = -x\) and therefore \(f(x) = -1\).

Step 2: Calculate the Right-Hand Limit at 0

The right-hand limit is examined as \(x\) approaches \(0\) from the positive side. Since \(f(x) = 1\) for all \(x > 0\), the right-hand limit as \(x \to 0^+\) is:\[ \lim_{{x \to 0^+}} f(x) = 1 \]

Step 3: Calculate the Left-Hand Limit at 0

The left-hand limit is examined as \(x\) approaches \(0\) from the negative side. Since \(f(x) = -1\) for all \(x < 0\), the left-hand limit as \(x \to 0^-\) is:\[ \lim_{{x \to 0^-}} f(x) = -1 \]

Step 4: Compare Right and Left Limits

The right-hand limit as \(x \to 0^+\) is 1, and the left-hand limit as \(x \to 0^-\) is -1. Since these two limits are not equal, \(f(x)\) has a jump discontinuity at \(x = 0\).

Step 5: Conclude on Continuity

Since the limits from the left and right at \(x = 0\) are not equal, and \(f(x)\) is not defined at \(x = 0\), the function \(f(x)\) is not continuous at \(0\). It has a jump discontinuity at this point.

Key Concepts

Discontinuity

In the world of calculus, a discontinuity is a point at which a function is not continuous. When we say a function is continuous at a point, it means that the graph of the function has no breaks, jumps, or holes there.
For a more mathematical definition, a function is continuous at a point if the following three conditions are met:

• The function is defined at that point.
• The limit of the function as it approaches the point from both the left and right is equal.
• The value of the function at that point and the limit are the same.

However, if any of these conditions fail, then the function is considered discontinuous at that point. Discontinuities can be due to a "jump" where the left-hand and right-hand limits don't match. Our given function, \( f(x) = \frac{|x|}{x} \), is a classic example because it is undefined at \( x = 0 \) and has different limits as you approach from the left and the right.
This discontinuity occurs because for values just above 0, \( f(x) \) repeatedly outputs 1, while for values just below 0, \( f(x) \) outputs -1. Therefore, we encounter a jump discontinuity at \( x = 0 \).

Right-Hand Limit

The right-hand limit refers to analyzing the behavior of a function as the variable approaches a specific value from the positive side, or from the right. Using notation, this is expressed as \( \lim_{{x \to a^+}} f(x) \). It tells us what the function values are getting closer to as \( x \) gets nearer to \( a \) but stays greater than \( a \).
For the function \( f(x) = \frac{|x|}{x} \), when we are looking at \( x \to 0^+ \), we only consider values of \( x \) that are positive. In this scenario, \( |x| = x \), making \( f(x) = 1 \). Thus, as \( x \) approaches 0 from the positive side, the function consistently equals 1, hence \[ \lim_{{x \to 0^+}} f(x) = 1.\]
This consistent behavior establishes the right-hand limit, helping us understand how the function behaves just before reacting to the point of discontinuity.

Left-Hand Limit

The left-hand limit is similar to the right-hand limit, only that we consider the approach from the negative side, also known as from the left. We express it as \( \lim_{{x \to a^-}} f(x) \), and it's about analyzing the values that the function approaches as \( x \) nears but is still less than \( a \).
When dealing with our specific function \( f(x) = \frac{|x|}{x} \), as \( x \to 0^- \), we focus on values where \( x \) is negative. Here, \( |x| = -x \), which makes \( f(x) = -1 \). Therefore, as \( x \) nears 0 from the left, \( f(x) \) continually outputs -1. This results in:\[ \lim_{{x \to 0^-}} f(x) = -1.\]
The left-hand limit provides insight into the function's behavior just at the cusp of the discontinuity, reinforcing the concept that at \( x = 0 \), the function behaves differently depending on the direction from which you approach.