Problem 67 - 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? $$ \begin{array}{|c|c|}\hline x & {1 / x} \\ \begin{array}{r}{1} \\ {2} \\\ {10} \\ {100} \\ {1000}\end{array} \\ \hline\end{array} $$ $$ \begin{array}{|c|c|}\hline x & {1 / x} \\ \hline 1.0 & {} \\ {0.5} & {} \\\ {0.1} & {} \\ {0.01} & {} \\ {0.001} & {} \\ \hline\end{array} $$

Step-by-Step Solution

Answer

As \(x\) increases, \(1/x\) approaches 0; as \(x\) decreases, \(1/x\) increases towards infinity.

Step 1: Calculate Reciprocals for Large x-values

Let's find the reciprocals of large values of \(x\) from the first table:- \(x = 1\), \(\frac{1}{x} = \frac{1}{1} = 1\)- \(x = 2\), \(\frac{1}{x} = \frac{1}{2} = 0.5\)- \(x = 10\), \(\frac{1}{x} = \frac{1}{10} = 0.1\)- \(x = 100\), \(\frac{1}{x} = \frac{1}{100} = 0.01\)- \(x = 1000\), \(\frac{1}{x} = \frac{1}{1000} = 0.001\)

Step 2: Observe the Pattern as x Increases

From the calculations in Step 1, notice that as \(x\) becomes larger, \(\frac{1}{x}\) becomes smaller. This pattern shows that as \(x\) approaches infinity, \(\frac{1}{x}\) approaches 0.

Step 3: Calculate Reciprocals for Small x-values

Now, let's find the reciprocals for smaller values of \(x\) from the second table:- \(x = 1.0\), \(\frac{1}{x} = 1\)- \(x = 0.5\), \(\frac{1}{x} = 2\)- \(x = 0.1\), \(\frac{1}{x} = 10\)- \(x = 0.01\), \(\frac{1}{x} = 100\)- \(x = 0.001\), \(\frac{1}{x} = 1000\)

Step 4: Observe the Pattern as x Decreases

From Step 3, observe that as \(x\) becomes smaller, approaching zero from the positive side, \(\frac{1}{x}\) becomes larger, increasing towards infinity.

Key Concepts

Understanding Reciprocals

To grasp the concept of reciprocals, imagine a simple fraction with one on top. The formula for a reciprocal is \(\frac{1}{x}\), where \(x\) is some number. This means you divide 1 by \(x\), transforming it into its opposite or reciprocal. It's like flipping a number around, creating its mirror image in terms of division. Here’s a quick example: if \(x = 2\), then \(\frac{1}{x} = 0.5\). This happens because \(2 \cdot 0.5 = 1\). It’s important to note that every number except zero has a reciprocal; zero is special and doesn’t have one as you could never divide by it (division by zero is undefined).

• Reciprocals transform multiplication into division.
• Every number except zero has a reciprocal.
• Reciprocals help in understanding inverse relationships.

Asymptotic Behavior of Reciprocal Functions

The asymptotic behavior of reciprocal functions is a fascinating part of their story. In simple terms, it describes how the function behaves as \(x\) becomes very large or very small. Specifically, for a function like \(\frac{1}{x}\), this term helps define what happens as the input values increase or decrease significantly.When \(x\) increases, \(\frac{1}{x}\) approaches zero, as we observed in the exercise's solution. Mathematically, this means that the graph of \(\frac{1}{x}\) gets very close to the x-axis but never actually touches it. This is known as a horizontal asymptote.

• When \(x\) is large, \(\frac{1}{x}\) nearly equals zero.
• The line \(y = 0\) acts as a horizontal asymptote.
• As \(x\) decreases to zero, \(\frac{1}{x}\) tends to infinity, displaying vertical asymptotic behavior.

Exploring Infinity in Mathematics

Infinity in mathematics represents an idea rather than a number. It denotes a concept of unboundedness or limitless quantity. When dealing with reciprocals, the term infinity becomes crucial. As \(x\) increases, \(\frac{1}{x}\) approaches zero, a behavior gradually leading it to an asymptotic zero without actually becoming zero. This is sometimes perceived as \(x\) approaching infinity.However, when \(x\) gets very small and gets closer to zero, \(\frac{1}{x}\) grows without bounds, suggesting infinity. This demonstrates a vital mathematical duality, where \(\frac{1}{x}\) heads towards zero or infinity based on \(x\)'s path, defining core aspects of calculus and analysis.

• Infinity is used to discuss potential limitless growth or decay.
• It's a key concept in comprehending asymptotic behavior of functions.
• Infinity provides a boundary point for unbounded sequences or functions.