Question

Interpret \(|f(x)-L|<\varepsilon\) in words.

Short answer

Answer: The inequality \(|f(x)-L|<\varepsilon\) represents that as the input value \(x\) gets closer to a specific value, the function \(f(x)\) approaches the limit \(L\) so closely that the distance between them is less than an arbitrarily small, positive number \(\varepsilon\). This helps us understand the concept of limits in calculus, where a function gets increasingly close to a certain value as the input approaches a particular point.

Step-by-step solution

Identify Symbols

There are three main symbols in this inequality: \(f(x)\), \(L\), and \(\varepsilon\). \(f(x)\) represents the value of the function at some point \(x\). In other words, this tells us what the function is outputting when we input \(x\). \(L\) is a constant representing the limit of the function. This means that as \(x\) gets really close to a specific value, the function \(f(x)\) gets really close to the value \(L\). \(\varepsilon\) is a small positive number, representing the margin of error or the "closeness" between the function and the limit. When we say that the function gets "really close" to the limit, this is our way of quantifying it.

Analyze the Inequality

The inequality states that \(|f(x)-L|\) is less than \(\varepsilon\). The absolute value symbol, \(| \cdot |\), is used to denote the distance between two values, meaning that this represents the distance between \(f(x)\) and \(L\). When we say that the distance is less than \(\varepsilon\), we're saying that the function's value (\(f(x)\)) is within \(\varepsilon\) units away from \(L\), but not equal to \(L\).

Interpret the Inequality in Words

Interpreting the inequality \(|f(x)-L|<\varepsilon\) in words would be: "The distance between the function's value at some point \(x\) and the limit \(L\) is less than a small positive number \(\varepsilon\)". This implies that as \(x\) gets closer to a specific value, the function \(f(x)\) approaches the limit \(L\) so closely that their difference (or distance) is less than an arbitrarily small, positive number \(\varepsilon\).

Key concepts

Epsilon-Delta Definition

The Epsilon-Delta definition is a formal way to understand the concept of limits in calculus. It essentially provides a rigorous framework for determining how a function behaves as it approaches a certain point. Here's a simple breakdown:

• Epsilon ( \(\varepsilon\) ): This represents how close we want the function's value to get to the actual limit. Think of it as a tiny "error margin," specifying that the function value should not stray more than this small distance from the limit.
• Delta ( \(\delta\) ): This symbolizes the proximity of the input values to the point of interest. When we say "for all values of \(x\) within \(\delta\) of a particular number," we mean that \(x\) must be within a narrow interval around this number.

The essence of this definition is captured in: "For every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that whenever the input \(x\) is within \(\delta\) units of a particular value, the output \(f(x)\) is within \(\varepsilon\) units of the limit \(L\)."
This essentially assures us that no matter how tightly we want to confine the function's outputs to the limit, we can always find an input interval that satisfies this condition.

Function Limit

A function limit describes the behavior of a function as the input approaches a certain value. When we say a limit exists, we mean that as our input \(x\) gets closer and closer to some number, the output \(f(x)\) approaches a specific value, known as the limit, \(L\).
Here's a simple way to understand limits:- Imagine driving towards a destination. As you get closer, you can roughly predict the time of arrival even before you reach it.- Similarly, in limits, as \(x\) nears a particular point, \(f(x)\) nears the limit without having to reach it necessarily.Limits are foundational in mathematics, especially for derivatives and integrals, helping us understand rates of change and areas under curves. They give us a snapshot of the function's behavior in snapshots right before and after a certain point. Importantly, the "approach" refers to the process, rather than arriving at the point itself.

Absolute Value in Inequalities

Absolute value is a concept that measures the "distance" between numbers on a number line, irrespective of direction. When dealing with inequalities involving absolute values, we're interested in how far apart two values are.Take the expression \(|f(x)-L|<\varepsilon\) as an example:- The absolute value \(|f(x) - L|\) indicates the distance between \(f(x)\) and the limit \(L\). - By stating that this distance is less than \(\varepsilon\), we express that \(f(x)\) is "close enough" to \(L\).This concept is vital because it eliminates any restriction on direction; it doesn't matter whether \(f(x)\) is above or below \(L\), just how close it is. Understanding this notion is key when applying the Epsilon-Delta definition and when solving real-world problems that necessitate knowing just the size of an error or difference, not whether it's positive or negative.
This "closeness without concern for direction" also corroborates that small perturbations in inputs result in small variations in outputs, which is critical in calculus.