Question

We say that \(\lim _{x \rightarrow \infty} f(x)=\infty\) 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. $$\lim _{x \rightarrow \infty} \frac{x}{100}=\infty$$

Short answer

Question: Prove that the limit of the given function as x approaches infinity is infinity. Function: \(f(x) = \frac{x}{100}\) Answer: Based on the steps provided above, it is proven that for any positive number M, there exists a corresponding positive number N (in this case, N = 100M) such that f(x) > M whenever x > N. Therefore, the limit of the function \(f(x) = \frac{x}{100}\) as x approaches infinity is infinity.

Step-by-step solution

Establish the inequality

We are given the functional form \(f(x) = \frac{x}{100}\). To prove the statement, we want to show that for any positive number \(M\), there exists a corresponding positive number \(N\) such that \(f(x) > M\) whenever \(x > N\).

Manipulate the inequality

To find the relationship between \(M\) and \(N\), we can work with the inequality we are trying to satisfy: \(f(x) > M\). Since \(f(x) = \frac{x}{100}\), we can write the inequality as: $$\frac{x}{100} > M$$

Solve for \(x\)

To find a relationship between \(M\) and \(N\) (where \(x > N\)), we need to solve the inequality for \(x\): $$x > 100M$$ Now, we see that if we choose \(N = 100M\), then whenever \(x > N\), we have \(x > 100M\), which means that our inequality holds, and thus our function satisfies the condition.

Conclude the proof

Since we found that when \(N = 100M\), the function \(f(x) = \frac{x}{100}\) satisfies the given definition of a limit going to infinity, we conclude that $$\lim_{x \rightarrow \infty} \frac{x}{100} = \infty$$

Key concepts

Inequality

In mathematics, inequalities form the basis of many proofs and problem-solving strategies. Here, an inequality is employed to demonstrate that a function diverges to infinity as its input grows without bounds. We start with an inequality derived from the functional form given:

• Set the inequality: \(f(x) = \frac{x}{100}\) should be greater than any prescribed positive number \(M\).
• The inequality becomes \(\frac{x}{100} > M\).

By solving this inequality, we find values of \(x\) (our input) that make it true. The goal is to make this inequality valid for all larger inputs, effectively showing that as \(x\) becomes larger than a certain threshold, \(f(x)\) surpasses any given value \(M\), hinting that \(f(x)\) grows without bounds.
Understanding this inequality is crucial as it helps establish whether the function meets the specific conditions needed to prove it diverges to infinity. It is a powerful tool for analyzing and predicting the behavior of functions at large values.

Function Growth

Functions can rise or fall rapidly with variations in their input, which is often described as their 'growth'. The concept of function growth is central to understanding limits at infinity, particularly when we deal with functions that become very large. For the function \(f(x) = \frac{x}{100}\), the growth is linear because as \(x\) increases, \(\frac{x}{100}\) grows at a steady rate. This consistent trend is easy to track:

• For any large enough \(x\), \(f(x)\) surpasses any set boundary \(M\), indicating persistent upward growth.
• The linear nature of \(\frac{x}{100}\) means that the larger \(x\) gets, the greater \(f(x)\) becomes, without approaching an upper limit.

Recognizing how function growth depends on its form can provide insights into finding limits and help anticipate long-term behavior of different functions. Hence, grasping function growth is as critical as solving specifics to resolve exercises on limits at infinity.

Proof Technique

The proof technique used in this exercise is based on manipulation and understanding of inequalities to assert that a function reaches infinity as its input goes to infinity. Let's break it down:Starting with a functional inequality as shown previously, the proof undergoes logical steps:

• Express the given condition (\( \frac{x}{100} > M \)) in terms of \(x\).
• Rearrange to isolate and solve for \(x\), hence determining suited \(N\): \(x > 100M\).
• Choose \(N = 100M\), making it a threshold beyond which the inequality holds.

Conclusively, for \(x > N\), \(f(x)\) becomes consistently greater than \(M\), which satisfies the definition of the limit approaching infinity. This structured approach leverages basic algebraic manipulations to methodically verify limit behaviors at infinity, providing a framework to validate such statements rigorously. A grasp of this proof technique is vital for tackling complex limits and verifying the divergence of functions to infinity.