Problem 101 - Essential Calculus

Question

In Exercises \(101-104,\) use the definition of limits at infinity to prove the limit. $$ \lim _{x \rightarrow \infty} \frac{1}{x^{2}}=0 $$

Step-by-Step Solution

Answer

By definition of limits at infinity, for every real number \(M > 0\), there exists a number \(N = \sqrt{\frac{1}{M}}\) such that whenever \(x > N\), \( \frac{1}{x^{2}} < M\). Therefore, \( \lim _{x \rightarrow \infty} \frac{1}{x^{2}} = 0 \).

Step 1: Approach

With the limit definition of infinity, for every real number M > 0, there exists a number N > 0 such that f(x) < M whenever x > N. Here, \(f(x) = \frac{1}{x^{2}}\), and we need to prove that for any \(M > 0\), we can find an \(N > 0\) such that when \(x > N\), \(f(x) < M\).

Step 2: Substitution

Substitute \(f(x)\) with \(M\) and solve for \(x\). This gives us \( M = \frac{1}{x^{2}} \), which simplifies to \( x^{2} = \frac{1}{M} \). Taking the square root of both sides gives \( x = \sqrt{\frac{1}{M}}\). This \(x\) is our \(N\).

Step 3: Proof

Given that \(M > 0\), it can be observed that the corresponding \(x = \sqrt{\frac{1}{M}}\) is always real and positive. Therefore, whenever \(x \) is more than \( \sqrt{\frac{1}{M}} \), the value of \( \frac{1}{x^{2}} \) will always be less than \(M\). This exactly fits the definition of the limit at infinity, so we can conclude that \( \lim _{x \rightarrow \infty} \frac{1}{x^{2}} = 0 \).

Key Concepts

Calculus

Calculus is a branch of mathematics that focuses on the study of change and motion. Understandably, it can be a complex subject, but when we break it down, it consists of two primal areas: differential calculus and integral calculus. Differential calculus concerns itself with the concept of a derivative, which essentially gives you the rate at which a quantity changes. On the other hand, integral calculus is about finding the quantity where the rate of change is known, which is called the integral.

Limits, such as the one in the exercise \( \lim _{x \rightarrow \infty} \frac{1}{x^{2}}=0 \), are a fundamental concept in calculus. These are used to describe the behavior of a function as the input approaches a certain value, which could be a real number or even infinity. Understanding limits is crucial because they form the foundation of both derivatives and integrals.

Proof of Limit

When we talk about a 'proof of limit,' we're referring to a logical argument that confirms a limit statement is true. This involves employing precise mathematical definitions and reasoning to demonstrate the exact conditions under which a limit statement holds.

For example, in the solution provided, the proof consists of a sequence of reasoned steps that use the definition of a limit at infinity to show that as \(x\) grows without bounds, \(\frac{1}{x^{2}}\) approaches zero. The proof is a critical piece to not just show that \( \lim _{x \rightarrow \infty} \frac{1}{x^{2}}=0 \) is true, but to understand why it's true. It requires an understanding of the structure of the mathematical theory and the ability to use logic and abstraction.

Limit Definition

The definition of a limit is a precise description of the behavior of a function as its argument gets close to a particular value or to infinity. In the case of the given exercise, the limit definition at infinity describes what happens to \( \frac{1}{x^{2}} \) as \( x \) becomes very large.

Specifically, the definition requires us to prove that for every \( M > 0 \) there exists an \( N \) such that for all \( x > N \), the function value is less than \( M \). Using the limit definition at infinity lays the groundwork for finding limits algebraically and is a fundamental skill for any student studying calculus.

Asymptotic Behavior

Asymptotic behavior in calculus refers to the way functions behave as they head towards infinity or towards a certain value. It gives us an understanding of the 'end behavior' of functions. In essence, it describes the tendencies of the function rather than exact values. The limit \( \lim _{x \rightarrow \infty} \frac{1}{x^{2}}=0 \) is an example of asymptotic behavior.

Here, as \( x \) increases without bound, the function \( \frac{1}{x^{2}} \) gets closer and closer to zero, meaning the function has an asymptote at \( y=0 \). This concept is crucial for graphing functions and analyzing their long-term behavior, which demonstrates not only how the function operates near the point of interest but also at the extreme ends of its domain.