Question

Absolute value limit Show that \(\lim _{x \rightarrow a}|x|=|a| .\) for any real number \(a .\) (Hint: Consider the cases \(a<0, a=0,\) and \(a>0 .\) )

Short answer

Question: Prove that for any real number \(a\), \(\lim_{x\rightarrow a}|x| = |a|\). Answer: We showed that for every real number \(a\), the limit as \(x\) approaches \(a\) of the absolute value of \(x\) equals the absolute value of \(a\) by considering three separate cases: when \(a<0\), when \(a=0\), and when \(a>0\). In each case, we considered sub-cases and showed that the limits exist, and they are equal to the absolute value of \(a\). Therefore, for any real number \(a\), \(\lim_{x\rightarrow a}|x| = |a|\).

Step-by-step solution

Case 1: \(a

When \(a<0\), we have two sub-cases: 1.1) When \(x < a\): In this case, both \(x\) and \(a\) are negative, so we have \(|x|=-x\) and \(|a|=-a\). Now let's find the limit as \(x\) approaches \(a\): \(\lim_{x\rightarrow a}|x| = \lim_{x\rightarrow a}(-x)\). Since \(-x\) is continuous for all real numbers, the limit exists: \(\lim_{x\rightarrow a}(-x) = -a = |a|\). 1.2) When \(x \ge a\): In this case, \(x\) is either negative and greater than \(a\) or \(x\) is nonnegative. In both cases, \(|x|=x\) and \(|a|=-a\). Thus, we need to show: \(\lim_{x\rightarrow a}|x| = \lim_{x\rightarrow a}(x)\). Since \(x\) is continuous for all real numbers, the limit exists: \(\lim_{x\rightarrow a}(x) = a = -(-a) = |a|\). Since both sub-cases lead to the same result, \(\lim_{x\rightarrow a}|x| = |a|\) when \(a<0\).

Case 2: \(a=0\)

When \(a=0\), we need to show that \(\lim_{x\rightarrow 0}|x| = |0|\). The function \(|x|\) is continuous for all real numbers, so the limit exists: \(\lim_{x\rightarrow 0}|x| = |0| = 0\).

Case 3: \(a>0\)

When \(a>0\), we have two sub-cases: 3.1) When \(x \le a\): In this case, \(x\) is either nonnegative and less than \(a\) or \(x\) is negative. In both cases, \(|x|=x\) and \(|a|=a\). Thus, we need to show: \(\lim_{x\rightarrow a}|x| = \lim_{x\rightarrow a}(x)\). Since \(x\) is continuous for all real numbers, the limit exists: \(\lim_{x\rightarrow a}(x) = a = |a|\). 3.2) When \(x > a\): In this case, both \(x\) and \(a\) are positive, so we have \(|x|=x\) and \(|a|=a\). Now let's find the limit as \(x\) approaches \(a\): \(\lim_{x\rightarrow a}|x| = \lim_{x\rightarrow a}(x)\). Since \(x\) is continuous for all real numbers, the limit exists: \(\lim_{x\rightarrow a}(x) = a = |a|\). Since both sub-cases lead to the same result, \(\lim_{x\rightarrow a}|x| = |a|\) when \(a>0\). In conclusion, for every real number \(a\), we have shown that \(\lim_{x\rightarrow a}|x| = |a|\).

Key concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It is divided into two main parts: differential calculus and integral calculus. Differential calculus concerns the determination of how functions change, which is captured by the concept of the derivative. Integral calculus, on the other hand, involves the accumulation of quantities, leading to the integral. A fundamental concept in calculus is the limit, which is essential for defining both the derivative and the integral.

For example, in exploring the behavior of the absolute value function as the input approaches a particular value, a limit helps us understand this behavior without explicitly reaching that specific point. This concept is crucial when dealing with continuous functions, as it provides a rigorous way to describe the function's behavior near points of interest, whether they are within the function's domain or at its boundaries.

Continuity

Continuity in mathematics describes a function that is unbroken or intact. A function is considered continuous at a point if the limit of the function as it approaches the point from both directions is equal to the function's value at that point. One intuitive aspect of continuity is that you can draw the graph of a continuous function without lifting your pen from the paper.

In the context of limits and absolute value, continuity plays a pivotal role. The function |x|, which gives us the absolute value, is continuous everywhere on the real number line. Because of this property, we can easily say that as x approaches any real number a, the limit of |x| will be |a|. In essence, continuity ensures that there are no surprises in the behavior of the function around the point of interest.

Limit of a Function

The limit of a function as the variable approaches a certain value is a fundamental concept in calculus. It describes the behavior of a function when the input gets arbitrarily close to a particular number. Limits can be used to understand the behavior of functions at points where they may not be directly defined, or where they exhibit sudden changes.

To illustrate, consider the absolute value function and its limit as x approaches any real number a. Whether a is negative, zero, or positive, we break the problem into different cases to assess the behavior of |x| as x approaches a from either side. This step-by-step approach, considering different intervals around a, allows us to capture the correct limiting value of |x|, which is |a| regardless of the situation, affirming the function's continuity and the existence of the limit in all cases.