Problem 14 - Precalculus: Mathematics for Calculus

Question

Estimating Limits Numerically and Graphically Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 0} \frac{\sqrt{x+9}-3}{x}$$

Step-by-Step Solution

Answer

The limit is approximately \(\frac{1}{6}\).

Step 1: Set Up the Function and Identify the Limit

We are given the function \(f(x) = \frac{\sqrt{x+9} - 3}{x}\) and need to estimate \(\lim_{x \to 0} f(x)\). This function becomes undefined at \(x = 0\), so we specifically need to explore its behavior as \(x\) approaches zero.

Step 2: Choose Values for Table

Select values of \(x\) that are close to zero, both negative and positive. This could include values such as \(-0.1, -0.01, -0.001, 0.001, 0.01, 0.1\). These will help us see the trend as \(x\) approaches zero.

Step 3: Calculate Values for the Table

For each chosen value of \(x\), compute \(f(x) = \frac{\sqrt{x+9} - 3}{x}\). For example:- If \(x = 0.1\), then \(f(0.1) = \frac{\sqrt{0.1+9} - 3}{0.1}\)- If \(x = -0.1\), then \(f(-0.1) = \frac{\sqrt{-0.1+9} - 3}{-0.1}\) Continue this process for each \(x\) value.

Step 4: Analyze the Table for a Limit

Examine the outputs from the table. As \(x\) approaches 0 from both the negative and positive sides, observe that the values of \(f(x)\) approach a particular number.

Step 5: Graphically Confirm the Limit

Use a graphing calculator or software to graph the function \(y = \frac{\sqrt{x+9} - 3}{x}\). Focus on the behavior of the graph around \(x = 0\). The limit is visually confirmed by the y-value that the graph approaches as \(x\) nears zero.

Key Concepts

Numerical Estimation

When dealing with limits, numerical estimation is an effective method to approximate the value of a limit by using a table of values. Let's look at the function \[ f(x) = \frac{\sqrt{x+9} - 3}{x} \]. This function becomes undefined at \( x = 0 \). Therefore, we estimate the limit as \( x \to 0 \).By choosing values of \( x \) close to zero, such as \(-0.1, -0.01, -0.001, 0.001, 0.01, 0.1\), we can observe how the function behaves. You calculate \( f(x) \) for each \( x \) in your table. For instance:

• If \( x = 0.1 \), then \( f(0.1) = \frac{\sqrt{0.1+9} - 3}{0.1} \).
• If \( x = -0.1 \), then \( f(-0.1) = \frac{\sqrt{-0.1+9} - 3}{-0.1} \).

Do this for each value of \( x \) near zero. Next, observe the trend of \( f(x) \) as \( x \) approaches zero from both sides.By analyzing these outputs, you may notice that the values are approaching a specific limit. This helps to numerically predict the value that \( f(x) \) is tending to, giving us the estimated limit.

Graphical Representation

Graphical representation of limits provides a visual way to confirm our numerical findings. By graphing the function \( y = \frac{\sqrt{x+9} - 3}{x} \), we can visually inspect the behavior near \( x = 0 \), where our function is undefined.Using a graphing calculator or software, plot the graph of the function over a small range around zero. This will allow you to see how the y-values change as \( x \) gets closer to zero. For instance:

• Focus especially on the portion where \( x \) is just slightly less than zero (left side) and slightly more than zero (right side).
• Observe the y-coordinates of the points on the graph as they get closer to \( x = 0 \).

You'll notice that even though the function is undefined exactly at \( x = 0 \), the y-values approach a particular number. This visual confirmation reassures you of the validity of your numerical estimation. Graphs are powerful tools to double-check the trends observed numerically.

Undefined Functions

In mathematical analysis, undefined functions occur often when evaluating limits. In our function \( f(x) = \frac{\sqrt{x+9} - 3}{x} \), it is undefined at \( x = 0 \). This occurs because substituting \( x = 0 \) into the denominator results in division by zero, which is not permissible.However, we can still evaluate the limit as \( x \) approaches this point, which is the essence of studying limits. Limits allow us to explore the behavior of functions near points where they are not defined. Here are tips to handle these functions:

• Use algebraic manipulation, if possible, to simplify the function. For some functions, this can eliminate the point of undefinedness.
• Use numerical methods like selecting values near the point in question to estimate trends.
• Graphically represent the function to gain insights into its behavior as it approaches the undefined point.

Understanding how functions behave at undefined points helps in comprehending various phenomena in calculus. It provides insight into continuity, limits, and the nature of functions beyond their standard definitions.