Problem 80 - Algebra and Trigonometry

Question

Limiting Behavior of Reciprocals Complete the tables. What happens to the size of the fraction 1\(/ x\) as \(x\) gets large? As \(x\) gets small?

Step-by-Step Solution

Answer

As \( x \to \infty \), \( \frac{1}{x} \to 0 \); as \( x \to 0^+ \), \( \frac{1}{x} \to \infty \); and as \( x \to 0^- \), \( \frac{1}{x} \to -\infty \).

Step 1: Understanding the Problem

We need to determine how the fraction \( \frac{1}{x} \) behaves as \( x \) becomes very large and as \( x \) becomes very small.

Step 2: Behavior as x Increases

As \( x \) increases, we can say that \( x \) is getting closer to infinity. Mathematically, when \( x \to \infty \), the fraction \( \frac{1}{x} \) approaches zero. This is because dividing a fixed number by an increasingly larger number yields a smaller result.

Step 3: Behavior as x Decreases

When we say \( x \) is decreasing in this context, we are considering \( x \) approaching zero from the left (i.e., from positive numbers to zero). As \( x \to 0^+ \), the fraction \( \frac{1}{x} \) grows very large. This is because dividing 1 by a number that's getting closer to zero gives a larger result.

Step 4: Behavior as x Approaches Zero from the Negative Side

Similarly, if \( x \) approaches zero from the negative side, \( x \to 0^- \), the fraction \( \frac{1}{x} \) becomes a very large negative number. This is the result of dividing 1 by a negative number that is close to zero.

Step 5: Conclusion

As \( x \) increases to positive infinity, \( \frac{1}{x} \) approaches zero. As \( x \) approaches zero from the positive side, \( \frac{1}{x} \) becomes vastly larger. Conversely, as \( x \to 0^- \), \( \frac{1}{x} \) becomes a large negative number.

Key Concepts

Limits

In mathematics, understanding limits is crucial, especially when analyzing the behavior of functions. The limit of a function describes the behavior of the function as its input approaches a particular point or as it goes to infinity. For the function \( f(x) = \frac{1}{x} \), examining its limit helps us comprehend what happens to the output as \( x \) changes.

When \( x \to \, \infty \), we look at what happens to \( \frac{1}{x} \). As \( x \) increases without bound, the value of \( \frac{1}{x} \) gets closer and closer to zero. Thus, we can write the limit as \( \lim_{{x \to \, \infty}} \frac{1}{x} = 0 \).

Similarly, understanding what happens as \( x \to 0^+ \) (approaching zero from the positive side) involves observing how the values become unbounded. The limit here can be described as \( \lim_{{x \to \, 0^+}} \frac{1}{x} = \, \infty \). This indicates an explosive increase in the value of the fraction. Limits provide a foundation for predicting how functions react under extreme conditions.

Behavior of Functions

Exploring the behavior of functions, particularly reciprocal ones like \( \frac{1}{x} \), reveals patterns that are central to calculus. Such behavior manifests differently when the input, \( x \), varies.

• As \( x \to \, \infty \): The function \( \frac{1}{x} \) trends towards zero. This pattern makes intuitive sense because a large denominator results in a smaller fraction.
• As \( x \to \, 0^+ \): The behavior diverges significantly, causing the function to shoot upwards towards positive infinity. The smaller \( x \) becomes, the larger \( \frac{1}{x} \) gets.
• As \( x \to \, 0^- \): The function acts similarly yet differently to the positive approach, resulting in a large negative value, revealing symmetry in reciprocal functions as they approach zero from either side.

These behaviors are instrumental in assessing how changes in the independent variable affect the dependent one. They are crucial for plotting graphs and modeling real-life phenomena that exhibit similar trends in behavior.

Infinity

Infinity is a concept in mathematics that goes beyond the finite and is necessary to describe the unbounded nature of some functions. Specifically, infinity is not a number but an idea that signifies something without limits.

In the realm of reciprocal functions, as \( x \) increasingly moves away from zero—becoming very large or very small—the output can become incredibly vast or infinitesimally minimal. For \( \frac{1}{x} \), when \( x \to \, \infty \), \( \frac{1}{x} \to \, 0 \), demonstrating that even as values become exceedingly large, the function never truly reaches zero, only approaches it indefinitely.

Conversely, as \( x \to \, 0 \), the function heads towards infinity, \( \frac{1}{x} \to \, \infty \). This concept helps explain how mathematical models can represent large, unpredictable increases or decreases, often seen in economics or physics. Embracing infinity allows us to grasp scenarios where no finite limit defines the scope of change.