Problem 2 - Basic Analysis: Introduction to Real Analysis

Question

Let \(f:[1, \infty) \rightarrow \mathbb{R}\) be a function. Define \(g:(0,1] \rightarrow \mathbb{R}\) via \(g(x):=f(1 / x) .\) Using the definitions of limits directly, show that \(\lim _{x \rightarrow 0^{+}} g(x)\) exists if and only if \(\lim _{x \rightarrow \infty} f(x)\) exists, in which case they are equal.

Step-by-Step Solution

Answer

The limits \(\lim_{x \rightarrow 0^{+}} g(x)\) and \(\lim_{x \rightarrow fty} f(x)\) exist and are equal when both exist.

Step 1: Interpret the Given Functions

We are given two functions: \(f:[1, \infty) \rightarrow \mathbb{R}\) and \(g:(0,1] \rightarrow \mathbb{R}\) where \(g(x) = f(1/x)\). The task is to understand how the behavior of \(f\) as \(x\) approaches infinity relates to the behavior of \(g\) as \(x\) approaches 0 from the positive side.

Step 2: Re-express the Function Limits

Rewrite \(g(x) = f(1/x)\). As \(x\) approaches 0 from the positive side, \(1/x\) approaches infinity. Therefore, \(\lim_{x \to 0^+} g(x) = \lim_{x \to 0^+} f(1/x) = \lim_{y \to \infty} f(y)\) by substituting \(y = 1/x\).

Step 3: Establish Conditions for Limit Existence

\(\lim_{x \to 0^+} g(x)\) exists if \(\lim_{y \to \infty} f(y)\) exists. This is because \(g(x)\) depends directly on \(f(1/x)\) where \(1/x\) increases without bound. Thus, the limit as \(y \to \infty\) determines if a finite limit exists for \(g(x)\).

Step 4: Show Equality of Limits

Assume \(\lim_{x \to 0^+} g(x) = L\). Then \(\lim_{y \to \infty} f(y) = L\) due to the substitution \(y = 1/x\). Conversely, if \(\lim_{y \to \infty} f(y) = L\), then \(\lim_{x \to 0^+} g(x) = L\). Thus, they must be equal when both exist.

Key Concepts

Real Analysis

In real analysis, we study the properties and behavior of real-valued functions and their limits. This field of mathematics helps us rigorously understand how functions behave under various transformations, especially as inputs approach specific values. We deal with concepts like continuity, smoothness, and convergence of sequences and functions.
When evaluating limits, such as in this exercise involving functions \(f\) and \(g\), we analyze how a function approaches a particular value as the input becomes infinitely large or infinitesimally small. Real analysis provides the rigorous foundation for these concepts, allowing us to derive meaningful results about function behaviors. In the exercise, we transform one limit condition into another using substitution, a technique deeply rooted in real analysis.

Function Behavior

The behavior of functions, specifically as they approach certain values, is key to understanding limits. In the discussed exercise, we examine two functions: \(f\) and \(g\), where \(g(x) = f(1/x)\). The behavior of \(g\) as \(x\) approaches \(0^+\) is directly linked to the behavior of \(f\) as \(x\) approaches infinity. This highlights how transformations can map one functional behavior into another.

• As \(x\) approaches \(0^+\), the value \(1/x\) becomes very large.
• This implies that to find out what \(g\) does near zero, we need to know how \(f\) behaves as it sees infinitely large inputs.

Understanding these transformations and behavior can often simplify complex problems into more manageable calculations, unveiling elegant relationships within mathematical structures.

Infinite Limits

Infinite limits describe the behavior of a function as its inputs become very large. Specifically, it refers to the limit of a function as the variable approaches infinity. This concept is pivotal when analyzing the asymptotic nature of functions. In this exercise, understanding infinite limits allows us to equate the limit of \(f(x)\) as \(x\) approaches infinity to the limit of \(g(x)\) as \(x\) approaches \(0^+\).
As \(x \to \infty\), the function \(f(x)\) may converge to a value, increase without bound, or oscillate indefinitely. Only if the first case occurs, do we state that \(\lim_{x \to \infty} f(x)\) exists, typically equating it to a real number. Alerts about the differences between the cases where a limit exists and does not exist can prevent common misunderstandings when dealing with infinite processes in calculus.

Limit Existence Conditions

Determining if a limit exists is fundamental when studying functions in calculus. This involves recognizing conditions under which a function approaches a specific value. For the given exercise, we need to show that the limit condition for \(g(x)\) exists if and only if it does for \(f(x)\), and further, they must equal when both limits exist.

• The limit \(\lim_{x \to 0^+} g(x)\) exists if \(\lim_{y \to \infty} f(y)\) does.
• This is due to how the limits are transposed through substitution \(y = 1/x\).

For proper existence, both conditions need to be finite and consistent. If a limit diverges, then it doesn't exist within the real number framework, thus emphasizing the importance of establishing finite limits. Only when both limits approach the same finite value under transformation can we confirm their equivalence, illustrating a unique and profound relationship between apparently different limit conditions.