Question

Find the limit \(L\). Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L\). $$ \lim _{x \rightarrow-2}|x-2| $$

Short answer

The limit \(L\) of the function as \(x\) approaches \(-2\) is 4. The epsilon-delta definition validates this by stablishing that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - -2| < \delta\), then \(||x - 2| - 4| < \varepsilon\). This indicates that the limit as \(x\) approaches \(-2\) is indeed 4.

Step-by-step solution

Identification of the limit

The limit of the function as \(x\) approaches \(-2\) is given by simply replacing \(x\) with \(-2\) in the function : \(|x - 2|\). Replacing \(x\) with \(-2\) in the function gives: \(|-2 - 2|\) which simplifies to 4. Thus, the limit \(L\) is 4.

Epsilon-Delta proof

After identifying the limit, the second step involves proving this limit using the epsilon-delta definition. This definition states that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - c| < \delta\), then \(|f(x) - L| < \varepsilon\). Here, \(L\) is the limit, which we have found to be 4, \(f(x) = |x - 2|\), and \(c\) is the point which \(x\) is approaching \(-2\). Hence, \(|f(x) - L| = ||x - 2| - 4|\). Now we need to determine \(\delta\) such that if \(|x - (-2)| < \delta\), then \(||x - 2| - 4| < \varepsilon\). Since \(|x - 2|\) is always positive, so \(||x - 2| - 4| = |4 - |x - 2||\). We also know \(|x | = | -2| = 2\) if \(|x + 2| = \delta\), meaning \(\delta = 2\). So for \(|x -2| - 4| < \varepsilon\), we can safely say that if \(\delta = \min{1, \varepsilon/2}\), then \(||x - 2| - 4| < \varepsilon\). Therefore, the limit is proven to be 4 using the epsilon-delta definition.

Key concepts

Limit of a Function

Understanding the concept of a limit is critical in calculus as it helps describe the behavior of functions as input values approach a certain point. In our exercise, we are tasked with finding the limit of the function as x approaches -2. Specifically, the function is the absolute value of (x - 2), denoted as \(|x - 2|\).

So, what does it mean to find the limit of this function as x approaches -2? It means that we're interested in predicting the value that \(|x - 2|\) gets closer to, as x becomes very close to -2 from either side. In our case, it turns out that this limit is 4.

Why is it important to grasp this concept? Because limits underpin many other concepts in calculus such as continuity and the derivative, which are fundamental for understanding how functions change and behave across their domains.

Absolute Value

The concept of absolute value represents the distance of a number from zero on a number line, regardless of direction. It's denoted by two vertical bars on either side of the number or expression, such as \(|x-2|\). The absolute value is always non-negative since it represents a distance.

So, when the exercise asked for the limit of \(|x-2|\) as x approaches -2, we were literally seeking how close the distance between x and 2 gets to a particular value when x is near -2. The absolute value of a real number x is defined as \(|x| = x\) if x is greater than or equal to 0, and \(|x| = -x\) if x is less than 0. This simple yet powerful concept is crucial in various areas of mathematics and its applications, particularly in measuring distances and defining the concept of magnitude.

Proof Techniques in Calculus

As we dive into proof techniques in calculus, the epsilon-delta definition stands out for its precision in defining limits. The epsilon-delta proof is often seen as a formal way of proving the limit of a function. It fundamentally states that for our limit to hold, the difference between the function's value and the limit (as x approaches a certain value) can be made as small as we like, provided we choose x close enough to that value.

Using our exercise, to properly execute an epsilon-delta proof, we need to find a suitable value of delta (\(\text{δ}\)) for every positive epsilon (\(\text{ε}\)). This involves manipulating the function in our case to find a condition where \(||x-2|-4|<\text{ε}\) occurs whenever \(|x -(-2)|<\text{δ}\). This process of setting up the appropriate inequalities and finding the concrete relationship between \(\text{ε}\) and \(\text{δ}\) reinforces one’s understanding of limits and builds a solid foundation for higher-level calculus problems and analysis.