Problem 81 - Thomas Calculus in SI Units

Question

a. Estimate the value of $$\lim _{x \rightarrow \infty}(x-\sqrt{x^{2}+x})$$ by graphing \(f(x)=x-\sqrt{x^{2}+x}\) over a suitably large interval of \(x\) -values. b. Now confirm your estimate by finding the limit with I'Hôpital's Rule. As the first step. multiply \(f(x)\) by the fraction \((x+\sqrt{x^{2}+x}) /(x+\sqrt{x^{2}+x})\) and simplify the new numerator.

Step-by-Step Solution

Answer

The limit is 0 as \(x\) approaches infinity.

Step 1: Understanding the Function

You need to find the limit \(\lim _{x \rightarrow \infty}(x-\sqrt{x^{2}+x})\). First, understand that you are subtracting the square root from \(x\), which may simplify using algebraic manipulation.

Step 2: Graphing the Function

Graph \(f(x) = x - \sqrt{x^2 + x}\) for a large range of \(x\) values, for example, from 1 to 10,000. Observe that as \(x\) becomes very large, \(f(x)\) appears to approach 0.

Step 3: Multply the Expression

Multiply \(f(x)\) by the fraction \(\frac{x + \sqrt{x^2 + x}}{x + \sqrt{x^2 + x}}\) to simplify the expression. This yields \(f(x) = \frac{(x - \sqrt{x^2 + x})(x + \sqrt{x^2 + x})}{x + \sqrt{x^2 + x}}\).

Step 4: Simplify the Numerator

Simplify the numerator: \((x - \sqrt{x^2 + x})(x + \sqrt{x^2 + x}) = x^2 - (x^2 + x) = x^2 - x^2 - x = -x\). Now, the expression becomes \(\frac{-x}{x + \sqrt{x^2 + x}}\).

Step 5: Applying L'Hôpital's Rule

To find the limit as \(x\) approaches infinity, use L'Hôpital's Rule. The function becomes indeterminate of the form \(-\frac{x}{2x + x^2} \approx -\frac{1}{2}\) as \(x\) approaches infinity. Differentiate numerator and denominator: numerator becomes \(-1\) and the derivative of the denominator is 1.

Step 6: Evaluate the Limit

The limit after differentiation using L'Hôpital's Rule is \(-\frac{1}{2}\). Therefore, \(\lim_{x \rightarrow \infty} \frac{-x}{x + \sqrt{x^2 + x}} = 0\), confirming the graphical observation.

Key Concepts

Graphical Analysis

Graphical analysis involves creating visual representations of functions to understand their behavior, especially in complex calculus problems. When dealing with limits, graphing the function can provide an intuitive look into how the function behaves as it approaches a specific value. In the exercise, the function \( f(x) = x - \sqrt{x^2 + x} \) is graphed to observe its trend as \( x \) increases towards infinity.
This visual summary allows us to identify that \( f(x) \), over large \( x \) values, seems to approach zero. Graphing over a large interval such as from 1 to 10,000 helps confirm that, despite its complex appearance, the function settles towards a particular value.
Key insights from graphical analysis include:

• Identifying patterns or trends.
• Confirming or predicting limit values.
• Visualizing complex relationships.

This method is crucial as it provides a straightforward, visual method to approach limits, acting as a first estimation before more rigorous techniques.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus to evaluate limits that initially present an indeterminate form. In our problem, the expression \( \lim_{x \rightarrow \infty} \frac{-x}{x + \sqrt{x^2 + x}} \) appears as an indeterminate form \( \frac{0}{\infty} \).
This rule provides a systematic way to solve such limits by differentiating the numerator and denominator separately, simplifying the process greatly. The differentiation is:

• Numerator: The derivative is \(-1\).
• Denominator: The derivative boils down to simplifying the terms to \(1\) if needed, or using more specific calculations depending on the complexity.

L'Hôpital's Rule helps us simplify the complex expression by focusing on derivatives, which are often simpler than the original expressions.
Once applied, we found that initially complicated limits are transformed into simpler expressions that clearly show that the intended limit approaches zero, which verifies our graphical estimation.

Algebraic Manipulation

Algebraic manipulation involves strategically rewriting expressions to simplify them. This is particularly useful in transforming complex expressions into manageable forms for further analysis. In this exercise, multiplying \( f(x) = x - \sqrt{x^2 + x} \) by \( \frac{x + \sqrt{x^2 + x}}{x + \sqrt{x^2 + x}} \) simplifies the problematic square root.
This manipulation exploits the difference of squares formula:

• \((a - b)(a + b) = a^2 - b^2\).

Applying it in the numerator: \((x - \sqrt{x^2 + x})(x + \sqrt{x^2 + x}) = x^2 - (x^2 + x) = -x\).
This nifty algebraic trick reduces mathematical complexity, turning a difficult analysis into a straightforward fraction \( \frac{-x}{x + \sqrt{x^2 + x}} \).
Thus, algebraic manipulation serves as an essential step to transform and simplify the function before applying other limit determination techniques like L'Hôpital’s Rule.