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Magazine Chapter 3: Problem 12
Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 1} \frac{1}{x}=1. $$
Short Answer
Expert verified
The limit is 1, proved by finding \( \delta = \varepsilon \) for \( \varepsilon > 0 \).
Step by step solution
01
Understand the Definition of Limit
The formal definition of the limit \( \lim_{x \to a} f(x) = L \) states 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 our case, \( a = 1 \), \( f(x) = \frac{1}{x} \), and \( L = 1 \). We need to show that for every \( \varepsilon > 0 \), a \( \delta > 0 \) can be found satisfying this criterion.
02
Set Up the Inequality for \( \varepsilon \)
We need to ensure that \( |f(x) - L| = \left| \frac{1}{x} - 1 \right| < \varepsilon \). Simplifying, we have \( \left| \frac{1 - x}{x} \right| < \varepsilon \). We aim to relate \( |1 - x| \) with \( \varepsilon \) by manipulating this inequality.
03
Simplify the Expression
We express \( \left| \frac{1 - x}{x} \right| < \varepsilon \) as \( \frac{|1 - x|}{|x|} < \varepsilon \). For \( x \) near 1, we choose \( \delta \) such that \( |x - 1| < \delta \) and therefore \( \delta \) must be small enough to also keep \( x \) close to 1 to ensure \( |x| \) stays bounded away from 0.
04
Establish the \( \delta \) Constraint
Ensuring \( \frac{|1-x|}{|x|} < \varepsilon \) means we need \( |1-x| < \varepsilon |x| \). Assume \( x \) is close enough to 1 so that \( |x| \approx 1 \). Find \( \delta \) as \( \delta = \frac{\varepsilon}{2} \). Thus, \( |1-x| < \delta \) ensures \( |f(x) - L| < \varepsilon \) by making \( \delta = \varepsilon |1| = \varepsilon \).
05
Finalize the Proof
Now, for \( 0 < |x - 1| < \delta = \varepsilon \), it follows that \( \left| \frac{1}{x} - 1 \right| < \varepsilon \). This concludes the proof, as we have shown \( \delta > 0 \) exists for given \( \varepsilon > 0 \). Hence, \( \lim_{x \to 1} \frac{1}{x} = 1 \) according to the formal definition of the limit.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Epsilon-Delta Definition
In calculus, the epsilon-delta (\(\varepsilon-\delta\)}) definition is a precise way to describe the behavior of functions as input values get infinitely close to a specific point. It forms the backbone for the formal definition of limits. Here鈥檚 how it works in simple terms:
- For every error margin (\(\varepsilon > 0\))}, which is how close you want to be to the limit, you can find a distance (\(\delta > 0\))}, which tells you how close the inputs need to be to a specific point.
- Whenever the input point x is within the distance \(0 < |x-a| < \delta\))}, the output value \(f(x)\))} will be within \(\varepsilon\))} of the limit.
Proof in Calculus
Crafting a proof in calculus usually involves establishing certain properties rigorously using definitions and logical arguments. The purpose is to demonstrate clearly that something is true based on fundamental principles. Here's a simple approach to understanding proofs using the epsilon-delta definition:First, identify the limit you want to prove. In our exercise, we aim to prove that \(\lim_{x \rightarrow 1} \frac{1}{x} = 1\))}. To do this, you:
- Assume any arbitrary small error margin, denoted by \(\varepsilon\))}. Your goal is to show that the difference between the function's value and the limit can be made smaller than \(\varepsilon\))}.
- Calculate or estimate how close \(x\))} needs to be to \(a\))} (the point x approaches), represented by \(\delta\))}, to ensure the outcome stays within the specified error margin (\(\varepsilon\))}).
- Use logical manipulation and arithmetic simplification to find a relationship between \(\varepsilon\))} and \(\delta\))}.
Limit of a Function
The concept of the limit of a function is central in calculus. It explains how a function behaves as it approaches a specific input value. Understanding limits helps in grasping continuous change and dealing with undefined operations. Let's break it down:
- A limit \(\lim_{x \to a} f(x) = L\))} implies that as \(x\))} gets closer and closer to \(a\))}, the function value \(f(x)\))} nears the value \(L\))}.
- This behavior helps us understand functions at points where they might not be directly measurable, such as points of discontinuity or where the function isn't explicitly defined.
- In practical terms, limits allow for the examination of trends and behaviors of functions over intervals and as they approach critical points.
