Problem 23 - Calculus

Question

Construct your own \(f(x)\) with these discontinuities at \(x=1\). Limit for \(x \rightarrow 1^{+},\) no limit for \(x \rightarrow 1^{-}\)

Step-by-Step Solution

Answer

Function: \( f(x) = \frac{1}{x-1} \) for \( x \leq 1 \) and \( f(x) = 2 \) for \( x > 1 \). Right limit exists, left does not.

Step 1: Understand the Requirements

To fulfill the requirements, we need to create a function \( f(x) \) with a discontinuity exactly at \( x = 1 \). Specifically, the function should have a limit from the right as \( x \to 1^+ \) and no limit from the left as \( x \to 1^- \).

Step 2: Design the Function Piecewise

Since we need different behaviors from the left and right of \( x = 1 \), constructing a piecewise function is ideal. For \( x > 1 \), choose a simple function like a constant function \( f(x) = 2 \) to ensure a limit exists as \( x \to 1^+ \). For \( x \leq 1 \), choose a function that oscillates or is undefined at \( x = 1 \), such as \( f(x) = \frac{1}{x-1} \).

Step 3: Define the Function

The function can be defined as: \[f(x) = \begin{cases} \frac{1}{x-1}, & \text{if } x \leq 1 \2, & \text{if } x > 1 \end{cases}\]In this piecewise function, as \( x \to 1^{+} \), the value of \( f(x) = 2 \), providing a clear limit from the right. As \( x \to 1^{-} \), \( f(x) = \frac{1}{x-1} \) becomes undefined.

Step 4: Verify the Right-Sided Limit

Evaluate the limit from the right. Consider \( \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} 2 = 2 \). The limit exists and equals 2.

Step 5: Verify the Left-Sided Behavior

Check the behavior from the left. As \( x \to 1^- \), \( f(x) = \frac{1}{x-1} \) approaches negative or positive infinity depending on the direction from which \( x \) approaches 1. Hence, no finite limit exists.

Key Concepts

Piecewise Function

A piecewise function is a type of function that is defined by multiple sub-functions, each of which applies to a specific interval of the domain. This means that the function can take different expressions based on the input value of the variable.
For example, consider the function we constructed: \[ f(x) = \begin{cases} \frac{1}{x-1}, & \text{if } x \leq 1 \ 2, & \text{if } x > 1 \end{cases} \]In this case, the function behaves differently depending on whether \(x\) is less than or equal to 1, or if \(x\) is greater than 1. Such a structure allows us to create specific behaviors around points of interest, such as discontinuities or abrupt changes in the function's values.
Piecewise functions are particularly useful in modeling real-world situations where a process might change behavior between different states or conditions.

Right-Hand Limit

The right-hand limit of a function at a specific point refers to the value that the function approaches as the variable gets close to that point from the positive side (i.e., values greater than the point). Essentially, it involves examining the function's behavior strictly from one side.
In our constructed piecewise function, we are interested in the right-hand limit at \(x = 1\).

• For \(x > 1\), \(f(x) = 2\), so \(\lim_{x \to 1^+} f(x) = 2\).

This indicates that as \(x\) approaches 1 from numbers greater than 1, the function value stabilizes at 2. The existence of this right-hand limit shows one type of behavior near the point of discontinuity.

Undefined Limit

An undefined limit at a point occurs when a function does not approach a specific value as the variable approaches that point. This can happen for several reasons, such as the function oscillating indefinitely or becoming unbounded.
In our exercise, the function \(f(x) = \frac{1}{x-1}\) for \(x \leq 1\) is of particular interest for values as \(x\) approaches 1 from the left. As \(x\) approaches 1 from the negative side (or numbers less than 1), the function's behavior becomes undefined because it leads to division by zero.

• As \(x\) nears 1 from the left, \(f(x)\) does not settle to a finite number; instead, it tends to either very large positive or negative values.

In other words, no finite limit exists for \(x \to 1^-\), which illustrates why we label it as 'undefined'.

Behavior of Functions Around Points

The behavior of functions around specific points is crucial for understanding their overall continuity and forms a foundation for many calculus concepts. When assessing this behavior, we often look at both limits and actual function values at and around the points of interest.
In the context of our function, the point \(x = 1\) is critical. This point represents where different behaviors from each piece of our piecewise function meet.

• For \(x \leq 1\): The function \(f(x) = \frac{1}{x-1}\) indicates drastic behavior changes as \(x\) nears 1.
• For \(x > 1\): The function \(f(x) = 2\) denotes consistent behavior leading to a constant limit.

Understanding this behavior allows for a deeper insight into the nature of discontinuities and the unique challenges they present in mathematical modeling and analysis.