Question

Define \((\mathbf{a}) \lim _{|\mathbf{X}| \rightarrow \infty} f(\mathbf{X})=\infty\) and \((\mathbf{b}) \lim _{|\mathbf{X}| \rightarrow \infty} f(\mathbf{X})=-\infty\).

Short answer

Question: Define the limit properties (a) \(lim_{|X| \rightarrow \infty} f(X) = \infty\) and (b) \(lim_{|X| \rightarrow \infty} f(X) = -\infty\), and describe the steps for evaluating these properties. Answer: The limit property (a) \(lim_{|X| \rightarrow \infty} f(X) = \infty\) implies that the output of function f(X) grows without bound as the magnitude of the input, |X|, approaches infinity. To evaluate this property, choose an arbitrary real number M, and find a real number N such that if |X| > N, then f(X) > M. If such a number N exists, the limit property holds. The limit property (b) \(lim_{|X| \rightarrow \infty} f(X) = -\infty\) implies that the output of function f(X) decreases without bound as the magnitude of the input, |X|, approaches infinity. To evaluate this property, choose an arbitrary real number M, and find a real number N such that if |X| > N, then f(X) < M. If such a number N exists, the limit property holds.

Step-by-step solution

Definition of (a) Limit Property: \(lim_{|X| \rightarrow \infty} f(X) = \infty\)

This property states that for a function \(f(X)\), the output, or value of the function, grows without bound (i.e., goes to infinity) as the magnitude of the input, \(|X|\), approaches infinity. In mathematical terms, it means that for any real number \(M\), there exists a real number \(N\) such that if \(|X| > N\), then \(f(X) > M\). This property is written as \((\mathbf{a}) \lim_{|X| \rightarrow \infty} f(X) = \infty\).

Steps for Evaluating (a) Limit Property

1. Choose an arbitrary real number \(M\). 2. Determine if there exists a real number \(N\) such that the inequality \(|X| > N\) implies \(f(X) > M\). 3. If such a number \(N\) exists, then the limit property holds, and we can say \((\mathbf{a}) \lim_{|X| \rightarrow \infty} f(X) = \infty\).

Definition of (b) Limit Property: \(lim_{|X| \rightarrow \infty} f(X) = -\infty\)

This property states that for a function \(f(X)\), the output or value of the function decreases without bound (i.e., goes to negative infinity) as the magnitude of the input, \(|X|\), approaches infinity. In mathematical terms, it means that for any real number \(M\), there exists a real number \(N\) such that if \(|X| > N\), then \(f(X) < M\). This property is written as \((\mathbf{b}) \lim_{|X| \rightarrow \infty} f(X) = -\infty\).

Steps for Evaluating (b) Limit Property

1. Choose an arbitrary real number \(M\). 2. Determine if there exists a real number \(N\) such that the inequality \(|X| > N\) implies \(f(X) < M\). 3. If such a number \(N\) exists, then the limit property holds, and we can say \((\mathbf{b}) \lim_{|X| \rightarrow \infty} f(X) = -\infty\).

Key concepts

Real Analysis

At the heart of calculus and higher mathematics lies real analysis, which provides a rigorous foundation for studying the properties of real numbers, sequences, functions, and limits. In essence, real analysis helps us to understand how mathematical functions behave and how they can be manipulated to deduce various properties.

Understanding limits, particularly the limits at infinity, is a fundamental concept within real analysis. For instance, when we say that \(\lim_{|X| \rightarrow \infty} f(X) = \infty\), what we really mean is that as the value of \(X\) grows larger and larger, the function \(f(X)\) increases without any bound. This kind of behavior is crucial in discerning the long-term behavior of functions, which can have implications in various fields such as physics, engineering, and economics.

It's also important to grasp the formal definition of limits at infinity, as provided in the step-by-step solution. This definition involves the concept of 'greater than any real number \(M\)', which allows mathematicians to express the idea of infinity in a precise way. By mastering the underlying principles of real analysis, students can develop a deeper comprehension of the mathematical landscapes they navigate.

Unbounded Behavior of Functions

The unbounded behavior of functions is an intriguing aspect of mathematics that deals with how functions react when their inputs become very large. In simpler terms, it's about what happens to a function's output when the input, denoted by \(X\) here, heads towards infinity. The behavior can be captured by the limit at infinity, described in two scenarios: either exploding to positive infinity or plummeting to negative infinity.

In case (a), \(\lim_{|X| \rightarrow \infty} f(X) = \infty\) signifies that the function \(f(X)\) grows without limitation as \(X\) increases. There is no ceiling to how high \(f(X)\) can go. Conversely, in case (b), \(\lim_{|X| \rightarrow \infty} f(X) = -\infty\) represents a function that falls without a floor as \(X\) increases. It keeps decreasing, without any bound on how low it can go.

Understanding and identifying this behavior is key in various applications. For example, in economics, the unbounded growth of a function could represent an unchecked increase in prices, leading to inflation. By evaluating these limits, one can predict such outcomes and plan accordingly.

Infinity in Calculus

Infinity in calculus is not just a number, it's a concept that represents boundlessness or endlessness. When we work with functions and limits, infinity allows us to describe the behavior of functions outside the constraints of finite values.

The idea of a function approaching infinity, as seen with \(\lim_{|X| \rightarrow \infty} f(X)\), is pivotal in calculus. It helps in understanding asymptotic behavior, where a function may get arbitrarily close to a line or value but never actually reach it. This brings a fascinating layer of complexity to function analysis, letting us predict and evaluate the long-term trends.

While infinity is not a real number that can be quantified, it serves as a powerful tool in calculus to express unbounded growth or decline. The steps outlined in solving the exercise offer an approach to determine whether, and how, a function exhibits such behavior. This understanding of infinity is not just a mathematical abstraction; it is widely applied in scientific fields to model real-world phenomena that grow or decay without limit.