Question

\(66-67 .\) Definition of infinite limits at infinity We write \(\lim _{x \rightarrow \infty} f(x)=\infty\) iffor 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. $$\lim _{x \rightarrow \infty} \frac{x}{100}=\infty$$

Short answer

Based on the provided step-by-step solution, create a short answer: To prove that the limit of \(\frac{x}{100}\) as x approaches infinity is indeed infinity, we used the definition of infinite limits at infinity. We let a positive M be chosen arbitrarily and aimed to find an N such that when x > N, the function f(x) = \(\frac{x}{100}\) will be greater than M. By choosing N = \(100M\), we were able to demonstrate that for any x > N, f(x) = \(\frac{x}{100}\) is greater than M. Therefore, according to the definition of infinite limits at infinity, we have proven that the limit \(\lim_{x \rightarrow \infty} \frac{x}{100} = \infty\).

Step-by-step solution

Assume a positive M

Let M be an arbitrarily chosen positive number. The goal is to find an N such that whenever x > N, the function f(x) = \(\frac{x}{100}\) will be greater than M.

Find an N that satisfies the condition

To find such an N, we need to show that: $$ \frac{x}{100} > M \quad \text{whenever} \quad x > N $$ Reorganizing this inequality, we have: $$ x > 100M $$ Thus, if we choose N = \(100M\), we have: $$ x > N $$

Show that the inequality holds for x > N

Now, let's show that the inequality holds for any x > N: 1. x > N 2. x > \(100M\) Since x > \(100M\), the function f(x) = \(\frac{x}{100}\) will result in f(x) > M.

Conclude the proof

We have shown that for any given positive M, we can find an N such that f(x) = \(\frac{x}{100}\) is greater than M whenever x > N. By the definition of infinite limits at infinity, this means that: $$ \lim_{x \rightarrow \infty} \frac{x}{100} = \infty $$ Thus, we have proven the statement using the definition of infinite limits at infinity.

Key concepts

Calculus

Calculus is a branch of mathematics that studies how things change. It's a tool that allows us to measure and analyze the dynamic world around us. The two main types of calculus are differential calculus, focusing on how things change over time, and integral calculus, which looks at the accumulation of quantities.

When we're faced with a function like \( f(x) = \frac{x}{100} \) in calculus, we're often interested in understanding its behavior as \( x \) becomes very large. This is where the concept of limits, particularly infinite limits at infinity, comes into play. By exploring how \( f(x) \) behaves as \( x \) approaches infinity, we gain insight into the growth behavior of the function, which is often critical in fields such as physics, engineering, and economics.

In this context, the step-by-step solution provided is an example of the application of limit laws and definitions to prove a specific type of limit statement, showcasing how calculus is not just about computation but also rigorous proof and logical reasoning. This foundation is indispensable for students aiming to navigate the more complex territories of mathematical analysis and applied science.

Limits of Functions

The concept of the limit of a function is essential in calculus, and it describes the behavior of a function at a certain point or as it approaches infinity. In our exercise, we're interested in infinite limits at infinity, specifically when the function \( f(x) \) becomes arbitrarily large as \( x \) itself becomes infinitely large. The formal definition used here serves as a gateway to understanding this behavior.

Using this definition, the solution involves showing that, for any arbitrarily large value of \( M \), there is some value of \( x \) from which point onward, the function will always be greater than \( M \). The implication of this proof is profound: it tells us that as we go further out along the \( x \) axis, the values of \( f(x) \) get larger without any bound. This concept is widely applicable, helping us to describe everything from exponentially growing populations to financial investments over time.

Understanding the mechanics of this proof equips students with the rigor to tackle more advanced problems and ensures they are not just performing rote calculations but also appreciating the underlying concepts of mathematical continuity and growth.

Proofs in Calculus

Proofs are the backbone of calculus and mathematics as a whole; they are logical arguments that establish the truth of mathematical statements. In calculus, proofs often hinge on the precise definition of concepts like limits, continuity, and derivatives.

As seen in the proof for the infinite limit at infinity, every step in the argument builds upon the previous one. By starting with the definition of infinite limits at infinity, we provide a basis for our reasoning. Then, by choosing an appropriate value of \( N \) that depends on \( M \) and showing that it satisfies the desired inequality, we craft a convincing argument that demonstrates the conclusiveness of \( \lim\text{ as }_{x \to \infty} \( \frac{x}{100} \) = \infty \).

The ability to prove such statements reinforces a student's understanding of calculus. Moreover, they gain not just the skill to solve specific problems, but also to communicate and justify their mathematical reasoning clearly and effectively. It's like learning a new language, one where each sentence must be accurate and precise to be valid, contributing to a greater appreciation and mastery of the subject.