Problem 85 - Calculus and its applications

Question

Find each limit. Use TABLE and start with \(\Delta\) Tbl \(=0.1\). Then use \(0.01,0.001,\) and \(0.0001 .\) When you think you know the limit, graph and use TRACE to verify your assertion. Then try to verify it algebraically. $$ \lim _{x \rightarrow 0} \frac{7-\sqrt{49-x^{2}}}{x} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Create a Table for Values of x

We'll start by making a table of values for \(x\) approaching 0 from both positive and negative sides. Begin with \(\Delta x = 0.1\) and then proceed with changes \(\Delta x = 0.01, 0.001,\) and \(0.0001\). Evaluate \( \frac{7-\sqrt{49-x^{2}}}{x} \) for each value. - For example: - When \(x = 0.1\), \(\frac{7-\sqrt{49-(0.1)^{2}}}{0.1}\) - When \(x = 0.01\), \(\frac{7-\sqrt{49-(0.01)^{2}}}{0.01}\) - And so forth.

Step 2: Analyze Table Results

Look for the trend in the computed values of the function as \(x\) approaches 0. Record the values and observe if they tend to some number. This helps predict the limit based on the pattern.

Step 3: Graph the Function

Plot the function \(y = \frac{7-\sqrt{49-x^{2}}}{x}\) using a graphing calculator or software. Use the TRACE feature to trace the function as \(x\) approaches 0 from both sides. Visually verify the behavior of the graph near \(x = 0\).

Step 4: Algebraically Simplify the Function

Since direct substitution leads to an indeterminate form, we can multiply the numerator and the denominator by the conjugate of the numerator: \[\frac{(7-\sqrt{49-x^{2}})(7+\sqrt{49-x^{2}})}{x(7+\sqrt{49-x^{2}})}\]Simplify the numerator using the difference of squares: \[(7-\sqrt{49-x^{2}})(7+\sqrt{49-x^{2}}) = (49 - (49-x^{2})) = x^{2}\]

Step 5: Final Algebraic Simplification

The expression now becomes: \[\frac{x^{2}}{x(7+\sqrt{49-x^{2}})} = \frac{x}{7+\sqrt{49-x^{2}}}\]Substitute \(x = 0\), which will lead to a definite number: \[\frac{0}{7+\sqrt{49-(0)}} = \frac{0}{14} = 0\] Thus, the limit is 0.

Key Concepts

Indeterminate Forms

When finding limits in calculus, you often encounter expressions that are seemingly undefined or indeterminate when evaluated directly. These are known as "indeterminate forms." A classic example of indeterminate form is \( \frac{0}{0} \). In our given problem, the direct substitution of \( x = 0 \) into \( \frac{7-\sqrt{49-x^{2}}}{x} \) results in \( \frac{0}{0} \). This tells us we can't simply substitute \( x \) for 0 in the expression.But don't worry! Indeterminate forms can often be resolved with clever algebraic manipulation, enabling us to determine the actual limit of the expression. Exploring these forms teaches us more about the function’s behavior near the point of indeterminacy, rather than directly at it.

Graphical Analysis

Graphical analysis is a powerful tool when trying to determine the behavior of a function near a specific point. By graphing the function \( y = \frac{7-\sqrt{49-x^{2}}}{x} \), we can visually inspect how the function behaves as \( x \) approaches 0. Using a graphing calculator or software, plot the function and use the TRACE feature. This helps to observe how the graph behaves as \( x \) edges closer to zero from both the positive and negative sides. As you trace the function, pay attention to any noticeable patterns or tendencies the graph exhibits near \( x = 0 \). In our solution, the graph visually indicated that the function approaches a limit value, reinforcing what we found in the table of values and pointing to further analytical exploration.

Algebraic Simplification

To solve indeterminate forms analytically, we often use algebraic simplification. The aim is to manipulate the function into a form that can be evaluated directly.In our problem, we use the algebraic trick of multiplying by the conjugate

1. Multiply both the numerator and the denominator by the conjugate of \( 7 - \sqrt{49 - x^2} \): \((7 + \sqrt{49 - x^2})\).
2. This step takes advantage of the difference of squares formula: \[(7 - \sqrt{49 - x^{2}})(7 + \sqrt{49 - x^{2}}) = 49 - (49 - x^2) = x^2\]
3. The resulting expression simplifies to: \[\frac{x^2}{x(7 + \sqrt{49-x^{2}})} = \frac{x}{7 + \sqrt{49-x^{2}}}\]

Now, substituting \( x = 0 \) yields a definitive result: \( \frac{0}{14} = 0 \). This algebraic manipulation confirms that the limit of the original expression is indeed 0, resolving the initial indeterminacy and validating our previous graphical and tabular findings.