Problem 20 - Precalculus Mathematics for Calculus

Question

Use a table of values to estimate the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow \infty}(\sqrt{9 x^{2}+x}-3 x)$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Set Up the Expression

We are given the limit \( \lim_{x \to \infty} (\sqrt{9x^2 + x} - 3x) \). Our goal is to estimate this limit using a table of values and verify it graphically.

Step 2: Simplify the Expression

Rewriting the expression inside the limit, \( \sqrt{9x^2 + x} - 3x \), we can factor out \( x \) from the inside of the square root: \( \sqrt{x^2(9 + \frac{1}{x})} - 3x \). This simplifies to \( x\sqrt{9 + \frac{1}{x}} - 3x \).

Step 3: Simplify Further Using a Common Factor

Factor \( x \) from both terms to get \( x(\sqrt{9 + \frac{1}{x}} - 3) \). Thus, the expression becomes \( x \left( \sqrt{9 + \frac{1}{x}} - 3 \right) \).

Step 4: Analyze the Limit

As \( x \to \infty \), the term \( \frac{1}{x} \to 0 \). Therefore, \( \sqrt{9 + \frac{1}{x}} \to \sqrt{9} = 3 \). This means the expression \( (\sqrt{9 + \frac{1}{x}} - 3) \to 0 \). Consequently, \( x(\sqrt{9 + \frac{1}{x}} - 3) \to x \times 0 = 0 \).

Step 5: Create a Table of Values

Select large values for \( x \) (e.g., 100, 1,000, 10,000) and calculate \( \sqrt{9x^2 + x} - 3x \) for these values. As \( x \) increases, observe whether the results approach 0.

Step 6: Graph the Expression

Using a graphing calculator or software, plot the function \( f(x) = \sqrt{9x^2 + x} - 3x \). Observe the behavior of \( f(x) \) as \( x \to \infty \). The graph should approach 0.

Key Concepts

Infinity

When we talk about the concept of infinity in mathematics, it can seem both vast and abstract. Infinity isn't a number that can be reached or a place you can get to—it’s an idea that describes something without any limit.
In calculus, especially concerning limits, we use infinity to describe the behavior of functions as the variable grows larger without bound. For example, when evaluating the limit \( \lim_{x \to \infty} (\sqrt{9x^2 + x} - 3x) \), we are interested in what happens to the expression as \( x \) progresses towards infinity.

• Growing Without Bound: As \( x \) increases, it moves towards infinity.
• Rate of Growth: Different expressions can grow at different rates.

Understanding infinity helps us predict the long-term behavior of expressions and functions. It's essential for making sense of how limits behave when variables take on infinitely large values.

Graphical Analysis

Graphical analysis is a powerful tool in mathematics that allows us to visually comprehend the behavior of functions. By plotting a graph, we can observe trends and get insights that might be less obvious just from the algebraic expression.
For instance, when we graph the function \( f(x) = \sqrt{9x^2 + x} - 3x \), we get a visual representation that helps us confirm our analytical findings.

• Behavior at Limits: The graph allows us to see what happens as \( x \to \infty \). For this function, the graph should approach zero.
• Visual Confirmation: Seeing that the function levels off at a particular value or approaches a limit graphically can solidify our analytical work.

The graphical analysis acts as a check. While the algebra gives us a precise definition, the graph can make the behavior of limits more intuitive and accessible.

Simplifying Expressions

In calculus and algebra, simplifying expressions is a vital skill that allows us to understand and solve problems more easily. Simplifying involves reducing an expression to its most basic form without changing its value.
For the expression \( \sqrt{9x^2 + x} - 3x \), simplifying it was key to finding the limit as \( x \to \infty \).

• Factoring and Cancelling: We factor out \( x \) inside the square root: \( \sqrt{x^2(9 + \frac{1}{x})} \), which simplifies to \( x\sqrt{9 + \frac{1}{x}} \).
• Converging Terms: As \( x \to \infty \), \( \frac{1}{x} \to 0 \), so \( \sqrt{9 + \frac{1}{x}} \to 3 \).

The expression then simplifies to \( x(\sqrt{9 + \frac{1}{x}} - 3) \), which leads to understanding that it approaches zero. Simplifying expressions makes it much easier to see the behavior of the function in the context of limits.