Question

Let \(\mathrm{A}\), be the set of points lying on the curve $$ y=\frac{1}{x^{\alpha}} \quad(0<\mathrm{x}<\infty) $$ What is $$ \bigcap_{\alpha \geqslant 1} A, ? $$

Short answer

The intersection \( \bigcap_{\alpha \geq 1} A \) is an empty set \( \emptyset \).

Step-by-step solution

Understanding the Intersection of Sets

To find \( \bigcap_{\alpha \geqslant 1} A \), we need to consider all sets \( A \) for each value of \( \alpha \) starting from \( \alpha = 1 \). Each \( A \) is defined as the set of points \((x, y)\) that satisfy \( y = \frac{1}{x^\alpha} \) for \( 0 < x < \infty \).

Evaluating the Condition for \( y \) with \( \alpha \geq 1 \)

For each \( \alpha \geq 1 \), \( y = \frac{1}{x^\alpha} \) represents a family of curves on the coordinate plane. As \( \alpha \) increases, \( \frac{1}{x^\alpha} \) decreases faster, essentially making the curve approach the x-axis for larger \( \alpha \).

Determining the Lowest \( y \)-value as \( \alpha \to \\infty \)

As \( \alpha \) increases further, every point \((x, y)\) requires that \( y \leq \frac{1}{x} \) since \( \frac{1}{x^\alpha} \leq \frac{1}{x} \) for all \( \alpha \geq 1 \). Thus, as \( \alpha \to \infty \), for any fixed \( x \), \( y \to 0 \).

Finding the Intersection of All Sets

The set \( \bigcap_{\alpha \geq 1} A \) must include only points that satisfy \( y = 0 \) for all \( \alpha \geq 1 \). This implies the intersection is the x-axis itself, but as \( y > 0 \) by definition, no points satisfy this condition except where the vertical line is tangently approaching zero as \( x \to \infty \).

Conclusion about the Intersection

The intersection \( \bigcap_{\alpha \geq 1} A \) covers no points meeting \( y = 0 \) based on all \( \alpha \), thereby resulting in an empty set. Thus, \( \bigcap_{\alpha \geq 1} A = \emptyset\).

Key concepts

Intersection of Sets

The concept of the intersection of sets is fundamental in set theory. When we talk about intersections, we refer to the common elements shared by two or more sets. In mathematical notation, the intersection is symbolized as \( \cap \). If we consider two sets, \( A \) and \( B \), their intersection \( A \cap B \) includes only the elements found in both \( A \) and \( B \).

In the context of the original exercise, we are interested in \( \bigcap_{\alpha \geqslant 1} A \), which involves an infinite collection of sets. Each set \( A \) corresponds to the points that lie on the curve described by the equation \( y = \frac{1}{x^\alpha} \). As \( \alpha \) increases, these curves draw nearer to the \( x \)-axis. Consequently, only points satisfying \( y = 0 \) for all \( \alpha \) remain common to all sets. However, since \( y > 0 \) by the given condition, the intersection becomes an empty set because no actual points will satisfy this criterion entirely.

Families of Curves

Families of curves offer a way to visualize functions as a collection of curves. By tweaking parameters such as \( \alpha \) in our function \( y = \frac{1}{x^\alpha} \), a family of curves is created on a graph. Each curve in this family represents a specific value of \( \alpha \).

As \( \alpha \) evolves, the curves shift in a manner that reflects changes in their geometric properties. Larger \( \alpha \) values lead to steeper declines as the curve hugs closer to the \( x \)-axis. This phenomenon is important in understanding limits because it illustrates how the positions and values of curves change as parameters are pushed to extreme values.

This understanding of the family of curves is critical when analyzing intersections and limits of functions, as it allows us to predict the behavior of a function under various conditions.

Limits of Functions

Limits are a crucial part of calculus and real analysis, providing insight into the behavior of functions as inputs approach certain values. To comprehend the limits of a function, we examine its output as the input approaches a particular point or extends infinitely in some direction.

In the provided exercise, limits help us understand what happens as \( \alpha \) increases indefinitely. For the function \( y = \frac{1}{x^\alpha} \), as \( \alpha \) becomes very large, the value of \( y \) tends towards zero for any fixed \( x \). This is because \( \frac{1}{x^\alpha} \) represents a progressively smaller number the larger \( \alpha \) gets.

Understanding limits is essential when determining intersections, as they help us define the exact nature of points (or lack thereof) shared across a family of curves. In this scenario, because \( y \leq \frac{1}{x} \) for all \( \alpha \geq 1 \), as \( \alpha \to \infty \), the limit forces all potential intersection points to shrink along the \( x \)-axis, resulting in an empty set.