Problem 15 - Precalculus: Mathematical for Calculus

Question

Use a table of values to estimate the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow-\infty} \frac{\sqrt{x^{2}+4 x}}{4 x+1}$$

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{4}\).

Step 1: Choose Values of x

To estimate the limit as \(x\) approaches negative infinity, select increasingly negative values for \(x\). For example, choose \(x = -10\), \(x = -100\), and \(x = -1000\). These values will help us observe the behavior of the function.

Step 2: Evaluate the Expression

For each chosen value of \(x\), compute \(\frac{\sqrt{x^{2}+4x}}{4x+1}\). This will give us an idea of how the function behaves as \(x\) approaches negative infinity.

Step 3: Analyze the Results

Examine the results from Step 2. If the values of the expression stabilize or approach a specific number as \(x\) becomes more negative, this number is approximately the limit.

Step 4: Algebraic Simplification

Simplify the expression \(\frac{\sqrt{x^{2}+4x}}{4x+1}\) for large \(x\). Factor out \(x\) from the square root: \(\frac{x\sqrt{1 + \frac{4}{x}}}{4x+1}\). As \(x\) approaches negative infinity, simplify as: \(\frac{-x\sqrt{1 + \frac{4}{x}}}{4x+1} = \frac{-x}{4x} = -\frac{1}{4}\).

Step 5: Confirm with a Graphing Device

Use a graphing tool to plot the function \(y = \frac{\sqrt{x^{2}+4x}}{4x+1}\) over a range of negative \(x\) values. Observe the graph to confirm if it visually approaches the value of \(-\frac{1}{4}\) as \(x\) approaches negative infinity.

Key Concepts

Infinite Limits

When we talk about infinite limits, we are exploring how functions behave as they reach extremely large positive or negative values. Here, understanding the function's behavior at extreme ends of the number line is crucial. As \( x \) approaches negative infinity, the table of values provides an estimation of the limit. Try these steps:

• Select values of \( x \) like \(-10, -100, -1000\), gradually moving to the left on the number line.
• Evaluate each value in the function \( \frac{\sqrt{x^{2}+4x}}{4x+1} \).
• Notice how the results behave. Are they tending towards a specific number?

This process helps us see the tendency of the function as \( x \) becomes very large in the negative direction.

Algebraic Simplification

Algebraic simplification is a method to make functions easier to analyze, especially as variables approach infinity. With this exercise, simplifying \( \frac{\sqrt{x^{2}+4x}}{4x+1} \) can make evaluating the limit more straightforward.Here's how we do it:

• Factor an \( x \) from the square root to simplify: \( \frac{x \sqrt{1 + \frac{4}{x}}}{4x+1} \).
• As \( x \) approaches negative infinity, observe how \( \frac{4}{x} \) tends to zero. This simplifies the square root to \( 1 \).
• Further simplify the expression to \( \frac{-x}{4x} \), which equals \(-\frac{1}{4} \).

This simplification provides a clearer picture of the function's behavior as \( x \) grows larger negatively.

Graphical Confirmation

Graphical confirmation is about visually verifying the conclusions reached through calculations. After algebraic simplification, using a graphing device helps in confirming the estimated limit. Follow these steps:

• Plot the function \( y = \frac{\sqrt{x^{2}+4x}}{4x+1} \) over a wide range of negative \( x \) values.
• Observe the graph. See if the curve approaches \(-\frac{1}{4} \) as \( x \) reduces to negative infinity.

Using both analytical and graphical methods provides a robust approach. This confirms that our mathematical simplifications and calculations are consistent with the graph. Graphical tools make abstract concepts more tangible, aiding deeper understanding.