Question

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

Short answer

Question: Show that the limit of the absolute value function at any point a is equal to the absolute value of the point a. Answer: We showed that in both cases (when a < 0 and when a ≥ 0), for any ε > 0, there exists a δ > 0 such that if 0 < |x-a| < δ, then ||x| - |a|| < ε. Therefore, the limit of the absolute value function at any point a is equal to the absolute value of the point a, or lim (x→a) |x| = |a|.

Step-by-step solution

Case 1 - When a < 0

Since \(a < 0\), it's clear that \(-a > 0\). The definition of the limit states that for each \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0<|x-a|<\delta\), then \(||x| - |-a|| < \epsilon\). Now, we can choose \(\delta = \epsilon\).

Observe the absolute value of the difference

Consider the expression \(||x| - |-a||\). We can rewrite this as \(||x| - |a||\). Now, think about the cases when \(x < 0\) and \(x \geq 0\). When \(x<0\), the expression \(|x - a| < \delta\) can be written as \(|-x+a| < \epsilon\). Since \(a < 0\) and \(x < 0\), we have \(-x+a = -x - (-a) = -x + a\). Thus, \(|-x - a| < \epsilon\), and it follows that \(||x| - |a|| < \epsilon\). When \(x \geq 0\), the expression \(|x - a| < \delta\) can be written as \(|x+a| < \epsilon\). Since \(a < 0\) and \(x \geq 0\), we have \(x+a = x - (-a) = x + a\). Thus, \(|x + a| < \epsilon\), and it follows that \(||x| - |a|| < \epsilon\).

Case 2 - When a ≥ 0

Since \(a \geq 0\), we use the limit definition again: for each \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0<|x-a|<\delta\), then \(||x| - |a|| < \epsilon\). We can choose \(\delta = \epsilon\) again.

Observe the absolute value of the difference in this case

When \(x \geq 0\), the expression \(|x - a| < \delta\) can be written as \(|x-a| < \epsilon\). Since \(a \geq 0\) and \(x \geq 0\), we have \(x - a = x - a\). Thus, \(|x - a| < \epsilon\), and it follows that \(||x| - |a|| < \epsilon\). When \(x < 0\), the expression \(|x - a| < \delta\) can be written as \(|-x+a| < \epsilon\). This time, \(a \geq 0\), and \(x < 0\), so we have \((-x) + a = a - x = a - (-x) = -a + x\). Thus, \(|-a + x| < \epsilon\), and it follows that \(||x| - |a|| < \epsilon\).

Conclusion

In both cases (\(a < 0\) and \(a \geq 0\)), we have shown that for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x-a| < \delta\), then \(||x| - |a|| < \epsilon\). Therefore, \(\lim_{x \rightarrow a} |x| = |a|\) for any real number a.

Key concepts

Absolute Value

The concept of absolute value is fundamental in mathematics. It represents the distance of a number from zero on a number line, regardless of direction. Whether positive or negative, the absolute value is always non-negative.
For a real number \(x\):

• If \(x \geq 0\), then \(|x| = x\).
• If \(x < 0\), then \(|x| = -x\).

Absolute value is crucial in limit problems and other mathematical concepts, as it allows us to focus on magnitude rather than direction. This neutrality is essential when dealing with equations like \(\lim_{x \rightarrow a}|x|=|a|\). By understanding how absolute values work, you can simplify expressions and make complex limit problems much easier to handle.

Epsilon-Delta Definition

The epsilon-delta definition is how mathematicians rigorously define limits. It's like a formal handshake between a function and its limit.
Here's how it works:

• For every \(\epsilon > 0\) (a small distance you choose), there exists a \(\delta > 0\) (a distance around \(a\)) such that \(0 < |x - a| < \delta\) ensures \(|f(x) - L| < \epsilon\).

In simpler terms, you can get \(f(x)\) as close as you want to \(L\) by making \(x\) sufficiently close to \(a\) without being equal to \(a\). This method is powerful for proving mathematical statements, as shown in demonstrating the limit of absolute value expressions. The epsilon-delta approach provides clarity in situations where intuition might fall short.

Real Numbers

Real numbers are the backbone of everyday mathematics. They encompass all the numbers that can be found on the number line, including rational numbers (like fractions) and irrational numbers (like \(\sqrt{2}\)).
Here’s a quick breakdown:

• Rational Numbers: Numbers that can be expressed as the fraction of two integers (e.g., \(\frac{1}{2}\), \(3\)).
• Irrational Numbers: Numbers that cannot be expressed as such fractions (e.g., \(\pi\), \(\sqrt{2}\)).

Real numbers play a vital role in calculus and analysis. They provide the setting for limits, like those involving absolute values. Understanding real numbers help you grasp how limits operate across different cases, whether \(a < 0\) or \(a \geq 0\). This breadth of scope is what makes real numbers so indispensable in mathematics.