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Chapter 2: Problem 23

Prove the statement using the \(\varepsilon, \delta\) definition of a limit. $$ \lim _{x \rightarrow a} x=a $$

Short Answer

Expert verified
Set \(\delta = \varepsilon\); this satisfies the limit definition.

Step by step solution

01

Understanding the Limit Definition

To prove the limit \[\lim_{x \to a} x = a\]using the \(\varepsilon, \delta\) definition, recall that for a function \(f(x)\), \(\lim_{x \to a} f(x) = L\) means that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that whenever \(0 < |x - a| < \delta\), it follows that \(|f(x) - L| < \varepsilon\). In this problem, we have \(f(x) = x\) and \(L = a\).
02

Applying the Limit Definition

In our specific case, we need to show that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - a| < \delta\), then \(|x - a| < \varepsilon\).
03

Choosing \(\delta\)

Notice that \(|x - a| < \varepsilon\) directly suggests that \(\delta = \varepsilon\) will work. We claim that for each \(\varepsilon > 0\), setting \(\delta = \varepsilon\) ensures that the inequality \(|x - a| < \varepsilon\) is true whenever \(0 < |x - a| < \delta\).
04

Confirming the Solution

Let's check: if \(0 < |x - a| < \delta = \varepsilon\), then it follows directly that \(|x - a| < \varepsilon\). This satisfies the \(\varepsilon, \delta\) definition required for the limit. Since every \(\varepsilon\) has such a corresponding \(\delta\), the limit \(\lim_{x \to a} x = a\) is proven.

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Key Concepts

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

Limit of a Function
The concept of the limit of a function helps us understand the behavior of functions as they approach a particular point. In calculus, a fundamental idea is that although we might not be able to plug in a number directly (often due to a function being undefined at that point), a function can still approach a particular value as it gets closer to that number.
This is where the formal definition of a limit comes into play. Understanding
  • When we say a function \(f(x)\) has a limit \(L\) as \(x\) approaches \(a\), it means the values of \(f(x)\) get closer and closer to \(L\) as \(x\) gets closer to \(a\).
  • Notation-wise, this is written as \(\lim_{x \to a} f(x) = L\).
In essence, limits help us formalize the intuition behind getting "infinitely close" to a point, allowing calculus to handle functions dynamically and rigorously.
Proof Techniques in Calculus
Proof techniques in calculus, like the \(\varepsilon, \delta\) definition of a limit, give us a structured way to prove that a function reaches a certain behavior as it approaches a point. The \(\varepsilon, \delta\) definition is a hallmark of mathematical rigor.The \(\varepsilon, \delta\) Method
  • This method provides a precise way to show that the limits exist by using two Greek letters: \(\varepsilon\) and \(\delta\).
  • It requires us to show that for any arbitrarily small positive number \(\varepsilon\), there is a corresponding positive number \(\delta\).
  • The condition \(0 < |x - a| < \delta\) directly implies \(|f(x) - L| < \varepsilon\).
For instance, proving \(\lim_{x \to a} x = a\) using this method involves:
  • Recognizing that our function is \(f(x) = x\) with \(L = a\).
  • Showing that \(|x - a| < \varepsilon\) naturally sets \(\delta = \varepsilon\), fulfilling the condition for any \(\varepsilon\) chosen.
This method solidifies our confidence in calculus: a field where precision and clarity rule.
Mathematical Notation
Mathematical notation is a precise language that allows us to communicate complex ideas succinctly and clearly. Understanding this notation is crucial in grasping limits and calculus as a whole.Common Symbols
  • \(\lim\): Denotes the limit, a fundamental concept in calculus signifying approaching values.
  • \(x \to a\): Represents "as \(x\) approaches \(a\)", showing the point of interest for the limit.
  • \(\varepsilon\) and \(\delta\): Greek letters representing positive quantities in proofs. They allow us to discuss the concepts of "arbitrary closeness" and "desired boundaries" respectively.
By internalizing the symbols and what they represent, one can navigate mathematical problems more effectively. The notation not only condenses information but also allows mathematicians to express ideas with unparalleled accuracy and consistency.

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Most popular questions from this chapter

Find the derivative of the function using the denition of derivative. State the domain of the function and the domain of its derivative. $$ f(t)=2.5 t^{2}+6 t $$

Find the numbers at which \(f\) is discontinuous. At which of these numbers is \(f\) continuous from the right, from the left, or neither? Sketch the graph of \(f\) $$ f(x)=\left\\{\begin{array}{ll}{2^{x}} & {\text { if } x \leqslant 1} \\ {3-x} & {\text { if } 14}\end{array}\right. $$

For what values of \(x\) is \(f\) continuous? $$ f(x)=\left\\{\begin{array}{ll}{0} & {\text { if } x \text { is rational }} \\\ {1} & {\text { if } x \text { is irrational }}\end{array}\right. $$

Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. $$ f(x)=\left\\{\begin{array}{ll}{x+3} & {\text { if } x \leqslant-1} \\ {2^{x}} & {\text { if } x>-1}\end{array} \quad a=-1\right. $$

For what value of the constant \(c\) is the function \(f\) continuous on \((-\infty, \infty) ?\) $$ f(x)=\left\\{\begin{array}{ll}{c x^{2}+2 x} & {\text { if } x<2} \\ {x^{3}-c x} & {\text { if } x \geqslant 2}\end{array}\right. $$
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