Problem 82 - 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. $$ \lim _{x \rightarrow 0} \frac{\sqrt{3-x}-\sqrt{3}}{x} $$

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{2\sqrt{3}}\).

Step 1: Set Up the Table with Initial Delta

Start by setting up a table of values for \( x \) using \( \Delta \text{Tbl} = 0.1 \). The values will be close to 0, such as \(-0.1, -0.05, 0.05, 0.1\). Evaluate the expression \( \frac{\sqrt{3-x}-\sqrt{3}}{x} \) for each of these values of \( x \).

Step 2: Refine the Table with Smaller Delta

Now, refine your table using \( \Delta \text{Tbl} = 0.01 \). Choose values such as \(-0.01, -0.005, 0.005, 0.01\) and evaluate the expression at each point. Repeat this process for \( \Delta \text{Tbl} = 0.001 \) and \( \Delta \text{Tbl} = 0.0001 \). Observe how the values change as you get closer to zero.

Step 3: Analyze the Trend

As the values of \( x \) get closer to zero, observe the output values of the table. Note the pattern as they approach a specific number, indicating the limit of the function. This is typically where all values stabilize to a single number.

Step 4: Graph and Use TRACE

Graph the function \( y = \frac{\sqrt{3-x}-\sqrt{3}}{x} \) using graphing software or a calculator. Use the TRACE feature to evaluate the function as \( x \) approaches 0. Compare these graphical insights with your table observations to make an assertion about the limit.

Step 5: Verify Algebraically

To verify algebraically, multiply the numerator and denominator by the conjugate, \( \sqrt{3-x} + \sqrt{3} \):\[ \frac{\sqrt{3-x}-\sqrt{3}}{x} \cdot \frac{\sqrt{3-x} + \sqrt{3}}{\sqrt{3-x} + \sqrt{3}} = \frac{(3-x) - 3}{x(\sqrt{3-x} + \sqrt{3})} \].This simplifies to: \[ \frac{-x}{x(\sqrt{3-x} + \sqrt{3})} = \frac{-1}{\sqrt{3-x} + \sqrt{3}} \].Take the limit as \( x \rightarrow 0 \), which gives \( \frac{-1}{2\sqrt{3}} \), thus confirming the solutions found through other methods.

Key Concepts

Numerical Methods

To find the limit, a common approach involves numerical methods, which can be quite effective when tackled methodically. The process starts by creating a table of values close to the point of interest—in this case, zero. By calculating the expression \( \frac{\sqrt{3-x}-\sqrt{3}}{x} \) at each value of \( x \), such as \(-0.1, -0.05, 0.05,\) and \(0.1\), you collect tangible data.

Refining the table by using smaller intervals like \( \Delta \text{Tbl} = 0.01 \), \(0.001\), and \(0.0001\) yields a more precise understanding of the behavior of the expression near zero.

• The outputs of the calculations provide a more apparent view of the pattern and how values stabilize or approach a single number as \( x \) nears zero.
• This pattern reveals the limit as \( x \) approaches zero.

Using this step-by-step numerical method allows you to predict the limit with increasing certainty and provides insight into the behavior of the expression around the point in question.

Algebraic Verification

Once a numerical estimate of the limit is achieved, algebraic verification provides confirmation. This involves manipulating the original expression to simplify or transform it into a form where the limit is more apparent. To address the indeterminate form presented as \( x \) approaches zero, multiply both numerator and denominator by the conjugate, \( \sqrt{3-x} + \sqrt{3} \).

This step utilizes a clever algebraic trick that exploits the difference of squares:

• The product \( (3-x) - 3 \) simplifies the numerator.
• Consequently, the expression becomes \( \frac{-1}{\sqrt{3-x} + \sqrt{3}} \).

Taking the limit of the simplified expression as \( x \rightarrow 0 \) is straightforward because the expression no longer has the indeterminate form of \( \frac{0}{0} \).

The limit can now be easily calculated, as it yields \( \frac{-1}{2\sqrt{3}} \) when \( x \) approaches zero, confirming the predictions from numerical exploration.

Graphical Verification

Visual representation through a graph provides a third form of verification, complementing both numerical and algebraic methods. By graphing the function \( y = \frac{\sqrt{3-x}-\sqrt{3}}{x} \), you obtain a visual depiction of its behavior as \( x \) approaches zero.

Using graphing software or a graphing calculator, follow these steps:

• Plot the graph to observe how the function behaves at values near zero.
• Utilize the TRACE feature to follow the function's value trajectory as \( x \) gets incrementally closer to zero.

The graph shows the trend visually, where the y-values begin to stabilize and converge towards \( \frac{-1}{2\sqrt{3}} \).

By cross-referencing this information with the numerical and algebraic insights, you gain a comprehensive understanding that confirms the limit from multiple perspectives. This multi-faceted approach ensures robust, verified conclusions for determining limits in calculus.