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Chapter 2: Problem 6

Sketch the interval \((a, b)\) on the \(x\) -axis with the point \(c\) inside. Then find a value of \(\delta > 0\) such that \(a < x < b\) whenever \(0<|x-c|<\delta\). $$a=2.7591, \quad b=3.2391, \quad c=3$$

Short Answer

Expert verified
Choose \(\delta = 0.2391\) to satisfy the conditions.

Step by step solution

01

Understand the Interval

We are given the interval \((a, b)\) where \(a = 2.7591\) and \(b = 3.2391\). The point \(c = 3\) lies within this interval because \(a < c < b\).
02

Draw the Interval on the x-axis

Sketch the number line and mark the points \(a = 2.7591\), \(b = 3.2391\), and \(c = 3\) on it. The interval \((a, b)\) is open, which means it includes all points between \(a\) and \(b\), but not \(a\) and \(b\) themselves. The point \(c = 3\) should be clearly marked inside this interval.
03

Understand the Condition |x-c| < 未

We need to find a positive \(\delta\) such that any \(x\) satisfying \(0 < |x-c| < \delta\) also satisfies \(a < x < b\). This means \(x\) must be inside the interval \((a, b)\), close to \(c = 3\).
04

Calculate 未 Based on the Interval

Since \(a < x < b\), we have two conditions: \(x < b\) translates to \(c + \delta < b\), and \(x > a\) translates to \(c - \delta > a\). Solving these give us: 1. \(c + \delta < b\) implies \(3 + \delta < 3.2391\) leads to \(\delta < 0.2391\).2. \(c - \delta > a\) implies \(3 - \delta > 2.7591\) leads to \(\delta < 0.2409\).The smaller value \(\delta = 0.2391\) will satisfy both conditions.
05

Conclude the Well-defined 未

Choose \(\delta = 0.2391\). This value ensures that \(x\) within \(\delta\) distance from \(c = 3\) is always inside the interval \((2.7591, 3.2391)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Intervals
In mathematics, an interval is a set of real numbers that includes all numbers between a pair of endpoints and possibly the endpoints themselves. For example, the interval \((a, b)\) discussed here is an *open interval*.
  • Open Interval: Only includes values between the endpoints \(a\) and \(b\), not \(a\) or \(b\) themselves.
  • Closed Interval: Includes the endpoints, noted as \([a, b]\).
  • Half-open Interval: Includes one endpoint, but not the other, like \([a, b)\) or \((a, b]\).
With an open interval \((a, b)\), any point \(x\) within it must satisfy the condition \(a < x < b\). This concept helps in understanding proximity within real numbers and is crucial for more complex topics like limits and continuity. Marking and visualizing these intervals on a number line provides clarity about what each type of interval includes.
Limits
The concept of limits is foundational in calculus. It describes the value that a function approaches as the input approaches some point. Limits are used to define derivatives, continuity, and integrals.
  • The expression \(\lim_{{x \to c}} f(x) = L\) means as \(x\) gets closer and closer to \(c\), \(f(x)\) approaches \(L\).
  • Limits allow us to analyze the behaviors of functions at points where they're not necessarily defined.
  • Used in defining derivatives, for instance \(f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h}.\)
When solving problems involving intervals and closeness, like our exercise, understanding limits helps us determine how values behave near specific points, even if they never actually reach the endpoints of intervals.
Epsilon-Delta Definition
The epsilon-delta definition is the formal way of describing limits and continuity rigorously.
  • Epsilon (蔚) represents how close the function value \(f(x)\) needs to be to \(L\), the expected limit value.
  • Delta (未) represents how close \(x\) needs to be to \(c\), except that \(xeq c\).
  • Formally, we say \(\lim_{{x \to c}} f(x) = L\) if for every \(\epsilon > 0\) there exists a \(\delta > 0\) so that whenever \(0 < |x - c| < \delta\), it follows that \(|f(x) - L| < \epsilon\).
Applying this to our exercise, we focused on finding a suitable \(\delta\) which ensures that any point within \( \delta \) of \(c = 3\) lies well within the interval \((a, b)\). The adjustments in 未 help accurately capture the behaviors of functions or intervals near specified points.
Real Analysis
Real Analysis is a branch of mathematics dealing with the set of real numbers and the functions defined on them. It provides deep insights into concepts such as convergence, limits, continuity, differentiation, and integration.
  • It involves rigorous proofs and definitions, moving beyond just intuitive ideas.
  • Studies sequences, series, and the behavior of functions using a more abstract approach.
  • Concepts like continuity and differentiability are core topics explored through precise definitions.
In our current context, real analysis provides the foundational tools for defining the interval's properties and writing limits formally. This discipline ensures that when mathematicians discuss notions like convergence or closed/open intervals, there is a concrete, methodical way to handle them.

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