Problem 38 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Question: Show that the limit of an absolute value function is equal to the absolute value of the limit. Answer: The limit of an absolute value function, \(\lim_{x \rightarrow a} |x|\), is equal to the absolute value of the limit, \(|a|\), for any real number. This is proven by considering two cases: when \(a<0\) (lim from left) and when \(a\geq0\) (lim from right), and showing that the limit for both cases is equal to \(|a|\).

Step 1: Case 1: a < 0

Since \(a < 0\), we know that \(|a| = -a\). We now need to show that \(\lim_{x \rightarrow a} |x| = -a\). When \(x < 0\) (i.e., as \(x\) approaches \(a\) from the left), we have \(|x| = -x\). Thus, \(\lim_{x \rightarrow a^-} |x| = \lim_{x \rightarrow a^-} (-x) = -a\).

Step 2: Case 2: a ≥ 0

Since \(a \geq 0\), we know that \(|a| = a\). We now need to show that \(\lim_{x \rightarrow a} |x| = a\). When \(x \geq 0\) (i.e., as \(x\) approaches \(a\) from the right), we have \(|x| = x\). Thus, \(\lim_{x \rightarrow a^+} |x| = \lim_{x \rightarrow a^+} x = a\).

Step 3: Limit for both cases

Since the limit from the left and the limit from the right are equal for both cases, we can conclude that \(\lim_{x \rightarrow a} |x| = |a|\), regardless of whether \(a<0\) or \(a \geq 0\). Therefore, \(\lim _{x \rightarrow a}|x|=|a|\) for any real number.

Key Concepts

Limits

Limits represent a fundamental concept in calculus and analysis, allowing us to understand how functions behave as they approach specific points or even infinity. When we talk about the limit of a function, we are essentially exploring what value the function nears as the input approaches a particular number. For example, with a function \(f(x)\), if \(\lim_{x \to a} f(x) = L\), it means as \(x\) approaches \(a\), \(f(x)\) gets very close to \(L\).

• Limits help in evaluating functions at points where they might otherwise be undefined.
• They are essential in defining other core concepts like continuity and derivatives.
• Techniques to evaluate limits include direct substitution, factoring, and rationalizing, among others.

Understanding limits is crucial when evaluating the behavior of functions involving absolute values, as these latter can have different expressions based on the input values, which is where the absolute value concept comes into play.

Absolute Value

The absolute value of a number represents its distance from zero on the number line, disregarding any direction. It's always non-negative. Thus, for any real number \(x\), the absolute value is expressed as \(|x|\), and it can be defined piecewise as:

• \(|x| = x\) when \(x \geq 0\)
• \(|x| = -x\) when \(x < 0\)

This definition helps to interpret \(|x|\) as turning any negative input into its positive counterpart. The challenge with limits involving absolute values often requires understanding these piecewise definitions, especially when the limit point could cause a change in the piece definition, like passing from negative to non-negative.
When determining the limit of an absolute value function as \(x\) approaches a point \(a\), you must consider both the left-hand and right-hand limits to confirm they are equal, as shown in the concept of one-sided limits.

One-Sided Limits

One-sided limits provide more detailed insights into the behavior of functions as they approach a particular point. They take into account the direction from which \(x\) approaches \(a\):

• The left-hand limit, denoted as \(\lim_{x\to a^-} f(x)\), considers values of \(x\) that are less than \(a\).
• The right-hand limit, \(\lim_{x\to a^+} f(x)\), considers values greater than \(a\).

For a full limit to exist at \(a\), both the left-hand limit and the right-hand limit must be equal. This is why one-sided limits are crucial when working with absolute values: depending on the sign of \(a\), the behavior of \(|x|\) changes, as does its analytic definition. By analyzing both \(x \to a^-\) and \(x \to a^+\), you ensure the limit \(\lim_{x \to a} |x|\) stabilizes, ultimately confirming it equates to \(|a|\). This approach underscores the critical importance of evaluating directions separately in calculus.