Problem 52 - Calculus Early Transcendentals

Question

We write \(\lim _{x \rightarrow \infty} f(x)=\infty\) if for any positive number \(M\) there is a corresponding \(N>0\) such that $$f(x)>M \quad \text { whenever } \quad x>N$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow \infty} \frac{x}{100}=\infty$$

Step-by-Step Solution

Answer

Question: Prove that the limit of the function \(f(x) = \frac{x}{100}\) tends to infinity as \(x\) tends to infinity. Answer: Based on the definition of the limit approaching infinity, we found a relationship between \(M\) and \(N\) such that for any positive number \(M\), we can choose \(N = 100M\), and when \(x > N\), we have \(f(x) > M\). Therefore, the limit of the function \(f(x) = \frac{x}{100}\) tends to infinity as \(x\) tends to infinity.

Step 1: Write down the definition of the limit

We are given that \(\lim _{x \rightarrow \infty} f(x) = \infty\), and the definition states that for any positive number \(M\), there exists an \(N > 0\) such that \(f(x) > M\) whenever \(x > N\). In this case, the function is given as \(f(x) = \frac{x}{100}\).

Step 2: Find a relationship between \(M\) and \(N\)

To find a relationship between \(M\) and \(N\), we will rewrite the inequality \(f(x) > M\) using the given function \(f(x) = \frac{x}{100}\). Rearranging the inequality, we have: $$\frac{x}{100} > M$$ $$x > 100M$$ Now we can see that for any \(M\) we choose, if we let \(N = 100M\), then the inequality \(x > 100M\) will be satisfied whenever \(x > N\).

Step 3: Prove the limit using the relationship between \(M\) and \(N\)

We will now prove that \(\lim _{x \rightarrow \infty} \frac{x}{100} = \infty\) using the relationship we found in step 2. Let \(M\) be any positive number. We will choose \(N = 100M\). Now, if \(x > N\), we have \(x > 100M\). Since the function \(f(x) = \frac{x}{100}\), when \(x > 100M\), we have: $$f(x) = \frac{x}{100} > \frac{100M}{100} = M$$ This shows that for any positive number \(M\), we can choose \(N = 100M\), such that \(f(x) > M\) whenever \(x > N\). Therefore, according to the definition, \(\lim _{x \rightarrow \infty} \frac{x}{100} = \infty\), which completes the proof.

Key Concepts

Limits at Infinity

Understanding limits at infinity is a pivotal part of calculus, especially when analyzing how functions behave as the input value grows without bound. The expression \( \lim_{x \rightarrow \infty} f(x) = \infty \) means that as \( x \) becomes larger and larger, the function \( f(x) \) increases without any upper boundary.

To visualize this concept, imagine shooting an arrow into the sky. With enough force, you expect it not to land, similar to a function's output growing indefinitely as \( x \) approaches infinity. In our exercise, we examined \( \lim_{x \rightarrow \infty} \frac{x}{100} = \infty \), confirming the idea that as \( x \) gets very large, dividing by a constant doesn't prevent \( f(x) \) from growing without bound.

Calculus

Calculus, the mathematical study of continuous change, is split into two main branches: differential calculus and integral calculus. In this context, we're dealing with a foundational concept in differential calculus—limits.

Without the concept of limits, we wouldn't be able to rigorously talk about instantaneous rates of change, which is the heart of derivative calculus. In the given problem, we apply the definition of limits to prove the behavior of a simple function as \( x \) increases without limitation. This grounding in limits ensures we have a well-defined language for the otherwise elusive concept of infinity in mathematical analysis.

Inequalities

Inequalities are statements that suggest a less than, greater than, or unequal relationship between two values or expressions. They are crucial when dealing with limit definitions involving infinity, as they provide a way to express that one quantity is larger than another beyond a certain point.

In the given exercise, the inequality \( \frac{x}{100} > M \) is used to establish a condition where the function \( f(x) \) exceeds any arbitrary positive value \( M \) we might choose. The solution shows that by appropriate selection of \( N \) (related to \( M \) by a clear inequality), we satisfy the limit definition. Inequalities are powerful tools in proving and understanding limit behaviors, essential in validating the limit definitions within calculus.

Proof Writing

In mathematics, proof writing is not only about showing that something is true, but also about communicating the reasoning clearly and logically. A well-constructed proof uses definitions, theorems, and logical deductions to demonstrate a proposition's truth.

When proving that \( \lim_{x \rightarrow \infty} \frac{x}{100} = \infty \) in the exercise, we presented a clear argument based on the formal definition of limits at infinity. The proof writing process included breaking down the problem into steps, choosing an appropriate value of \( N \) based on \( M \) to satisfy the conditions set by the limit definition, and showing the consistency of this choice regardless of how large \( M \) gets. Strong proof writing skills are essential in higher-level mathematics, as they provide a robust foundation for understanding complex concepts.