Problem 9 A positive number $\epsilon$ a... [FREE SOLUTION]

Chapter 2: Problem 9

A positive number $\epsilon$ and the limit $L$ of a function $f$ at a point $a$ are given. Find a number $\delta$ such that $|f(x)-L|<\epsilon$ if $0<|x-a|<\delta$ $$\lim _{r \rightarrow 2} \frac{x^{2}-4}{x-2}=4 ; \epsilon=0.05$$

Short Answer

Expert verified

$ \delta = 0.05 $

Step by step solution

Identify the Limit Statement

We need to prove that $ \lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}=4 $ using the definition of the limit. We have an $ \epsilon $ value of 0.05, meaning we need to find a $ \delta $ such that for all $ x $ meeting the condition $ 0<|x-2|<\delta $, the inequality $ |\frac{x^{2}-4}{x-2} - 4|< 0.05 $ holds.

Simplify the Expression

Simplify the expression $ \frac{x^{2}-4}{x-2} $. By factoring, we recognize the numerator as a difference of squares: $ x^2 - 4 = (x-2)(x+2) $. Thus the expression simplifies to $ x + 2 $ when $ x eq 2 $.

Set Up the Inequality

We transform the condition into an inequality: $ |x+2 - 4| < 0.05 $. Simplifying it gives $|x - 2| < 0.05 $. This represents how close we must keep $x$ to 2 to ensure $f(x)$ stays within 0.05 units of the limit 4.

Find the Suitable $\delta$

Since the inequality is $ |x - 2| < 0.05 $, we can directly use these bounds. Thus, we choose $ \delta = 0.05 $ to satisfy the condition $ 0<|x-2|<\delta $ implies $ |f(x) - 4|<0.05 $.

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Key Concepts

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

Epsilon-Delta Definition

The epsilon-delta definition is the rigorous foundation of what it means for a limit to exist in calculus. To understand this definition, it's helpful to realize that it formalizes the idea of getting arbitrarily close to a certain value. Here's how it works:

The function's limit as it approaches a point is represented as a certain number, say $L$.
If you want to say that "as $x$ approaches $a$, $f(x)$ gets closer to $L$," you pick a small positive number, $\epsilon$, which defines how close $f(x)$ needs to be to $L$.
You must find another small positive number, $\delta$, so that whenever $x$ is within $\delta$ of $a$ (except at $a$), $f(x)$ is within $\epsilon$ of $L$.

This formal definition helps ensure that the concept of limits is precise and universal, not relying on intuition or guesswork.

Limit Proof

To prove a limit, like the exercise provided, you employ the epsilon-delta definition. The aim is to demonstrate that for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x$ satisfying $0 < |x-a| < \delta$, the expression $|f(x) - L| < \epsilon$ holds true. This involves:

Expression Simplification: Start by simplifying the function's expression to make the problem more tractable.
Inequality Setup: Use the simplified expression to set up an inequality that captures the closeness of $f(x)$ to the limit $L$.
Choosing $\delta$: Adjust the inequality until you find a suitable $\delta$ that works for the given $\epsilon$.

Ultimately, proving a limit using the epsilon-delta method requires both algebraic manipulation and logical reasoning to establish that $f(x)$ can stay as close to $L$ as desired, by keeping $x$ close to $a$.

Difference of Squares

A common algebraic technique used in limit proofs is the difference of squares. This method simplifies expressions by recognizing patterns that can be factored:

The expression $x^2 - 4$ is a difference of squares because it can be rewritten as $(x-2)(x+2)$.
Factoring the expression can simplify the function, such as converting $\frac{x^2-4}{x-2}$ to $x + 2$ when $x eq 2$, making it easier to analyze around the point of interest.

Using the difference of squares is particularly helpful because it breaks down more complicated expressions and allows for straightforward simplification, especially when dealing with rational functions in limit calculations.

Inequality Manipulation

Manipulating inequalities is a critical skill in proving limits, particularly to confirm they fit within the bounds set by $\epsilon$ and $\delta$. Here's what you do:

Set Up: Begin with the inequality derived from your simplified expression, such as $|x + 2 - 4| < \epsilon$.
Adjust & Solve: Rearrange and solve the inequality to express $\epsilon$ in terms of $|x-a|$. In our given example, it became $|x - 2| < 0.05$.

By manipulating these inequalities, you determine the precise $\delta$ that ensures within that range of $x$, the function's value will not stray too far from the limit value, $L$, thus satisfying $\epsilon$. This logical consistency is fundamental to completing a limit proof.

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91影视

A positive number \(\epsilon\) and the limit \(L\) of a function \(f\) at a point \(a\) are given. Find a number \(\delta\) such that \(|f(x)-L|<\epsilon\) if \(0<|x-a|<\delta\) $$\lim _{r \rightarrow 2} \frac{x^{2}-4}{x-2}=4 ; \epsilon=0.05$$

Short Answer

Step by step solution

Identify the Limit Statement

Simplify the Expression

Set Up the Inequality

Find the Suitable \(\delta\)

Key Concepts

Epsilon-Delta Definition

Limit Proof

Difference of Squares

Inequality Manipulation

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