Problem 65 - Calculus: Early Transcendentals

Question

Definition of a limit at infinity The limit at infinity \(\lim _{x \rightarrow \infty} f(x)=L\) means that for any \(\varepsilon>0,\) there exists \(N>0\) such that $$|f(x)-L|<\varepsilon \text { whenever } x>N$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{2 x+1}{x}=2$$

Step-by-Step Solution

Answer

Question: Prove that \(\lim_{x \rightarrow \infty} \frac{2x+1}{x} = 2\) using the definition of limit at infinity. Answer: We showed that for any \(\varepsilon > 0\), we can choose \(N = \frac{1}{\varepsilon}\) such that \(\left|\frac{1}{x}\right| < \varepsilon\) whenever \(x > N\). Since this holds true for any positive \(\varepsilon\), we have proven that \(\lim_{x \to \infty} \frac{2x + 1}{x} = 2\).

Step 1: Simplify the expression

First, let's simplify the expression \(|\frac{2x+1}{x} - 2|\). By finding a common denominator, this can be rewritten as: $$\left|\frac{2x+1-2x}{x}\right|$$ which simplifies to: $$\left|\frac{1}{x}\right|$$

Step 2: Choose N based on ε

We need to find an \(N\) such that for all \(x > N\), the inequality \(|\frac{1}{x}| < \varepsilon\) holds. To do this, we can pick \(N = \frac{1}{\varepsilon}\), since if \(x > N\) then \(\frac{1}{x} < \varepsilon\) by definition.

Step 3: Prove the inequality

Let \(x > N = \frac{1}{\varepsilon}\). Then, we have \(\frac{1}{x} < \varepsilon\). Thus, $$\left|\frac{1}{x}\right| = \frac{1}{x} < \varepsilon$$Since this holds for any \(\varepsilon > 0\), we have shown that $$\lim_{x \to \infty} \frac{2x + 1}{x} = 2$$ by the definition of the limit at infinity.

Key Concepts

Limit of a Function

The concept of a limit is a fundamental building block in calculus, as it helps us understand the behavior of functions as they approach a certain point or infinity. Simply put, the limit of a function at a particular point describes the value that the function approaches as the input (or 'x' value) gets increasingly close to that point. When we say that the limit of a function as x approaches infinity equals 'L', we are saying that as x grows larger and larger without bound, the function's value gets closer and closer to 'L'.

Consider our exercise example: \(\lim _{x \rightarrow \infty} \frac{2x+1}{x}=2\). As x approaches infinity, the fraction \(\frac{2x+1}{x}\) gets closer and closer to 2, which suggests that the horizontal asymptote of this function is y=2. This intuitive understanding is substantiated through a formal proof using the epsilon-delta definition of a limit.

Epsilon-Delta Definition of a Limit

The epsilon-delta definition provides a precise and formal way to define what we mean by 'limits' in calculus. According to this definition, for a limit \(L\) at infinity, for every positive number \(\varepsilon\), there exists a corresponding number \(N\) such that for all x above N, the absolute difference between the function's value and L is less than \(\varepsilon\). This means we can make the function's value as close to L as we desire, by choosing x sufficiently large.

In the context of our exercise, we applied this definition to prove that \(\lim _{x \rightarrow \infty} \frac{2 x+1}{x}=2\). We pick an \(\varepsilon > 0\) and find a corresponding \(N\) (in this case, \(N = \frac{1}{\varepsilon}\)), which ensures that for all x greater than N, \(\left|\frac{2x+1}{x} - 2\right| < \varepsilon\) holds true. This formal method authoritatively concludes that the function approaches the value of 2 as x approaches infinity.

Simplifying Expressions

Simplifying expressions is a skill that makes complex calculus problems more manageable. By rewriting expressions in a simpler form, we can often gain insight into the behavior of a function and apply calculus concepts more easily. In our exercise, we simplified \(\left|\frac{2x+1}{x} - 2\right|\) by subtracting like terms and reducing the expression to \(\left|\frac{1}{x}\right|\), a much simpler form to work with.

Why Simplify?

Simplification aids in:

• Identifying patterns in expressions
• Making calculations easier
• Finding limits, derivatives, and integrals more straightforwardly

A well-simplified expression can significantly clarify the steps needed to solve a problem, as seen in our decision to make \(N = \frac{1}{\varepsilon}\) for proving the limit in question.

Calculus Proofs

Proofs in calculus are essential for verifying the true behavior of mathematical statements and functions. They provide a solid foundation to support conclusions drawn from our calculations and assumptions. In our exercise, proving that \(\lim _{x \rightarrow \infty} \frac{2 x+1}{x}=2\) involved logical reasoning and appropriate application of definitions and properties of limits.

The Role of Proofs

Proofs play a key role in calculus by:

• Ensuring that our solutions for equations and expressions have a rigorous basis
• Helping us explore and understand the properties of functions deeply
• Laying groundwork for more advanced mathematical theories

The proof for the given exercise uses a blend of simplification techniques and the epsilon-delta definition to establish a concise and irrefutable conclusion.