Question

Does the set \(\\{x: 0<|x-a|<\delta\\}\) include the point \(x=a ?\) Explain.

Short answer

Answer: No, the set {x: 0<|x-a|<δ} does not include the point x=a.

Step-by-step solution

Substitute the value of 'x' with 'a'

Let's substitute \(x=a\) in the given inequality: \(0<|a-a|<\delta\).

Simplify the inequality

Simplifying the inequality, we get: \(0<|0|<\delta\). Since \(|0|=0\), the inequality becomes: \(0<0<\delta\).

Analyze the inequality

As we can see, \(0<0\) is false since zero cannot be less than itself. Hence, the inequality \(0<|x-a|<\delta\) is not true when \(x=a\).

Conclusion

Since the inequality is not satisfied for \(x=a\), we conclude that the set \(\\{x: 0<|x-a|<\delta\\}\) does not include the point \(x=a\).

Key concepts

Inequality Simplification

Inequality simplification is an important mathematical skill that helps students understand and solve complex problems. Simplification often involves reducing an inequality to its most basic form to make it easier to analyze. In the context of the provided exercise, the process of simplification is evident. The student starts by substituting the variable 'x' with 'a', which, effectively reduces the inequality to an expression that can be easily evaluated: from \(0<|x-a|<\ta\) to \(0<|a-a|<\ta\), and eventually to \(0<0<\tta\).

At each step of simplification, the form becomes simpler until the inequality either holds true or is shown to be false. This step-by-step approach of breaking down the inequality is crucial in understanding at what point the inequality does not hold, which in the given problem is at the comparison of zero with itself. Simplifying inequalities requires careful attention to the rules of algebra and understanding of absolute values—a topic that we'll explore further in the next section.

Concept of Delta in Calculus

The concept of delta (\(\ta\)) in calculus is often used to describe a small, positive quantity that represents a difference or change. It is frequently used in the context of limits, continuity, and the formal definition of a derivative. The role of delta is to provide a precise measurement of how close two values are to each other—how one value can approach another within a given range or tolerance.

When tackling problems involving delta, such as those with absolute value inequalities, it is crucial to understand that delta represents the allowable margin of error around a certain point. In the given exercise, delta is used to create a range \((0, \tn\ta)\) around the point 'a'. This range dictates how close 'x' has to be to 'a' without actually being 'a' itself. This understanding is vital when studying concepts like limits and continuity, where delta helps to assert how functions behave near a specific point.

Absolute Value Properties

Absolute value is a fundamental concept in mathematics that describes the distance of a number from zero on the number line, irrespective of its sign. Symbolically, the absolute value of 'x' is denoted by \(|x|\), which is always a non-negative value. This property plays a significant role in solving inequalities because it encapsulates both the positive and negative possibilities of a variable.

Some critical properties of absolute values that come into play in inequality questions are that \(|x| = x\) if 'x' is positive or zero, and \(|x| = -x\) if 'x' is negative. In the context of the exercise, when the student evaluated \(|a-a|\), the property that the absolute value of zero is zero (\(|0| = 0\)) was applied, leading to the very simple inequality of \(0<0<\ta\) which is not possible, revealing that the point 'x=a' is not in the proposed range defined by \(0<|x-a|<\ta\). Understanding the precise behavior of absolute values is key to solving problems involving absolute value inequalities.