91Ó°ÊÓ

One-sided limits help us understand the behavior of functions as they get closer to a specific point from either side. Unlike a regular limit, which looks at both sides simultaneously, one-sided limits focus on just one direction:
- **Left-Hand Limit (LHL)**: This is denoted as \(\lim_{x \to a^-} f(x)\). It examines what happens to the value of the function as \(x\) approaches \(a\) from the left, meaning with values slightly smaller than \(a\).
- **Right-Hand Limit (RHL)**: This is symbolized by \(\lim_{x \to a^+} f(x)\). It checks the function's behavior as \(x\) comes close to \(a\) from the right, with values just larger than \(a\).
In our exercise, we calculated both of these limits for \(a=2\):
- As \(x\) approaches from the left (\

A function is said to be discontinuous at a point if it jumps, breaks, or has a hole at that point.
In terms of limits, discontinuity typically occurs when the two-sided limit doesn't exist or doesn't equal the function's value at that point.
**Types of Discontinuities:**

• Jump Discontinuity: Happens when the left-hand limit and right-hand limit exist but are not equal. This is exactly what we found in our exercise, with a jump from -1 to 1 at \(x=2\).


• Removable Discontinuity: This occurs when the limit exists but it's not equal to the function's value at that point.


• Infinite Discontinuity: Involves a limit that approaches infinity, creating a vertical asymptote.

These types help in categorizing how the function behaves around points of interest, particularly in piecewise functions where different rules apply for different intervals.

Piecewise functions are those defined by different expressions for different intervals of the domain. Instead of a single formula for all values, piecewise functions have multiple 'pieces' with specific rules.
**Why Piecewise Functions Matter:**

• They capture real-world situations: Many scenarios don't follow a single rule. For instance, tax brackets or shipping costs may follow different formulas for different income levels or weights.


• Useful for discontinuous functions: Functions with jumps or holes can be accurately described using multiple equations.

Our given function \(f(x) = \frac{x-2}{|x-2|}\) can be seen as piecewise because it acts differently when \(x\) is less than or greater than 2.
For \(x < 2\), it simplifies to \(-1\), and for \(x > 2\), it turns into \(1\). This separation into distinct behaviors at \(x=2\) highlights the utility of piecewise definitions.