Problem 20 - Precalculus: Mathematics for Calculus

Question

Estimating Limits Numerically and Graphically 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 approximately 0.5.

Step 1: Understand the Problem

We need to estimate the limit \( \lim_{x \to \infty}(\sqrt{9x^2+x}-3x) \). This involves determining the value that \( \sqrt{9x^2+x}-3x \) approaches as \( x \) becomes infinitely large.

Step 2: Set Up a Table of Values

Choose values of \( x \) that are sufficiently large to observe the behavior of the expression. For example, evaluate the expression at \( x = 100, 1000, 10000, \) and so on.

Step 3: Calculate Expression Values

For each chosen \( x \), compute \( \sqrt{9x^2+x} - 3x \). For example:- When \( x = 100 \), \( \sqrt{9(100)^2 + 100} \approx 301 \) and \( 3 \times 100 = 300 \), so the expression is \( \approx 1 \).- When \( x = 1000 \), \( \sqrt{9(1000)^2 + 1000} \approx 3001 \) and \( 3 \times 1000 = 3000 \), so the expression is \( \approx 1 \).- Continue for larger \( x \) values.

Step 4: Analyze the Numerical Limit

Notice that as \( x \) increases, the expression \( \sqrt{9x^2+x} - 3x \) seems to approach \( 0.5 \). The values are consistently getting closer to 0 as \( x \) becomes very large.

Step 5: Graphically Estimate the Limit

Use a graphing calculator or software to graph \( y = \sqrt{9x^2+x} - 3x \). Observe the behavior of the graph as \( x \) approaches infinity. The graph should show that the function values approach a horizontal line near zero.

Key Concepts

Numerical Estimation

Numerical estimation is a method to approximate a limit by evaluating the function's values at several points. When trying to estimate limits numerically, you typically create a table of values for the function. Select values of \( x \) that progress towards or beyond the point of interest, in this case towards infinity.

For the given problem, the expression is \( \sqrt{9x^2 + x} - 3x \). We start with large values of \( x \) like 100, 1000, and 10000 to see how the function behaves. By calculating the expression for each of these values, we notice a trend. For instance:
- At \( x = 100 \), the result is approximately 1.
- At \( x = 1000 \), the result is also around 1.
- With increasing \( x \), the output gets closer to 0.

It becomes clear as \( x \) grows larger, the value of \( \sqrt{9x^2 + x} - 3x \) approaches closer to 0, providing an empirical confirmation through numerical estimation.

Graphical Estimation

Graphical estimation involves using a graph to visualize how a function behaves as \( x \) approaches a limit. This method complements numerical estimation by providing a visual context.

For this problem, use a graphing calculator or software to plot \( y = \sqrt{9x^2 + x} - 3x \). Start by setting a suitable window that allows you to observe the function as \( x \) turns to infinity. Upon observing the graph, you would notice:
- The graph draws nearer to a horizontal line, which, in this case, is the x-axis.
- The function's values become nearly indistinguishable from zero as \( x \) increases.

This graphical method efficiently supports the numerical findings, providing a solid confirmation that the limit as \( x \) approaches infinity is indeed 0.

Infinite Limits

Infinite limits describe the behavior of a function as \( x \) either approaches infinity or a particular value that causes the function to soar to infinity. In higher mathematics, evaluating infinite limits is crucial for understanding asymptotic behavior.

In this exercise, we approach the specific case of \( \lim_{x \to \infty} (\sqrt{9x^2 + x} - 3x) \). As \( x \) becomes infinitely large, the behavior is determined by comparing the dominant terms within the expression. Here, \( \sqrt{9x^2} = 3x \) is the leading term.

Subtract \( 3x \) from \( \sqrt{9x^2 + x} \), leaving only the smaller parts of the equation, which diminish in comparison, hence pushing the overall value closer to zero. This results in the limit approaching zero as \( x \) tends towards infinity.

Understanding this helps students grasp how limits behave over unbounded intervals, a key concept especially in calculus where infinite processes are often involved.