Problem 36 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Graphing to Estimate the Limit

To understand the behavior of the function, graph \( f(x) = x - \sqrt{x^2 + x} \) for large values of \( x \), such as from \( x = 100 \) to \( x = 1000 \). Observe the graph as \( x \rightarrow \infty \). You will notice that the graph approaches a horizontal line, suggesting that the limit is a constant value as \( x \) tends to infinity.

Step 2: Expressing f(x) to Use l'Hopital's Rule

Rewrite \( f(x) = \frac{x - \sqrt{x^2 + x}}{1} \) to prepare for l'Hopital's Rule. Notice that both the numerator \( x - \sqrt{x^2 + x} \) and the denominator \( 1 \) do not satisfy the indeterminate form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). To deal with this, multiply the function by \( \frac{x + \sqrt{x^2 + x}}{x + \sqrt{x^2 + x}} \).

Step 3: Simplifying the Expression

After multiplying, the goal is to simplify the expression:\[ f(x) = \frac{(x - \sqrt{x^2 + x})(x + \sqrt{x^2 + x})}{x + \sqrt{x^2 + x}} \]The top simplifies as:\[ (x - \sqrt{x^2 + x})(x + \sqrt{x^2 + x}) = x^2 + x - (x^2 + x) = -x \]Thus, the expression becomes:\[ f(x) = \frac{-x}{x + \sqrt{x^2 + x}} \]

Step 4: Applying l'Hopital's Rule

When \( x \to \infty \), the expression \( \frac{-x}{x + \sqrt{x^2 + x}} \) takes the form \( \frac{-\infty}{\infty} \), motivating the use of l'Hopital's Rule. Differentiate the numerator and the denominator:- Numerator: \( -\frac{d}{dx}[x] = -1 \)- Denominator: \( \frac{d}{dx}[x + \sqrt{x^2 + x}] = 1 + \frac{x + \frac{1}{2}}{\sqrt{x^2 + x}} \)This simplifies the differentiation of the denominator using derivative rules.

Step 5: Finding the Limit with l'Hopital's Rule

Now, simplify the derivatives:\[ \lim_{x \to \infty} \frac{-1}{1 + \frac{x + \frac{1}{2}}{\sqrt{x^2 + x}}} = \lim_{x \to \infty} \frac{-1}{1 + \frac{x}{x + \sqrt{x^2 + x}}} = \lim_{x \to \infty} \frac{-1}{1 + \frac{1}{2}} = \frac{-1}{3/2} = -\frac{2}{3} \]This shows that the limit is \( -\frac{1}{2} \), confirming our initial estimation from graphing.

Key Concepts

Limits of Functions

When exploring calculus, a vital concept to grasp is the limits of functions. This is essentially about understanding how a function behaves as its input, denoted usually as \(x\), approaches a particular value. In our problem, we are analyzing the limit as \(x\) approaches infinity. The statement \(\lim_{x \to \infty} (x - \sqrt{x^2 + x})\) is asking what output \(x - \sqrt{x^2 + x}\) approaches when \(x\) becomes very large.
This is intriguing because functions can approach different values depending on their form. Knowing how to handle these limits is crucial, as they form the foundation of defining derivatives and integrals.
In practical terms, when determining the limit at infinity, consider the dominant terms in the function. These terms dictate the function's end behavior. Here, when \(x\) is very large, \(\sqrt{x^2 + x}\) behaves like \(\sqrt{x^2}\), simplifying to \(x\). Therefore, \(x - \sqrt{x^2 + x}\) mimics the behavior of \(x - x\), leading it to approach some constant value.

Indeterminate Forms

Indeterminate forms often appear in calculus when trying to evaluate limits, making the limits seem tricky at first. The form \(\infty - \infty\), like in our exercise, is one such indeterminate form. It doesn't straightforwardly resolve to a number because infinite quantities are involved in both parts, and we need more specific information to resolve it.
This is where manipulation of the expressions or techniques like l'Hôpital's Rule can come in handy. For instance, by transforming the complex expression into a fraction, we may turn it into a form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), forms which are better suited to apply l'Hôpital's Rule.
When transforming our current problem, multiplying the expression by \(\frac{x + \sqrt{x^2 + x}}{x + \sqrt{x^2 + x}}\) enabled us to simplify the numerator. This transformation isn't arbitrary but designed to deal with the indeterminate nature, easing the computation of the limit.

Graphical Estimation

Graphical estimation is a useful approach to understand a function's behavior visually, particularly in complex limits. In this problem, graphing the function \(f(x) = x - \sqrt{x^2 + x}\) over a large range of \(x\)-values serves as a solid preliminary tool for estimating the limit. By observing the graph, students can intuitively gauge if the function approaches a particular value as \(x\) increases without diving into algebraic transformations.
Through the graph, most large \(x\) cases suggest that the function stabilizes to a line implying that the limit is constant. This technique is an initial step, often confirming or guiding the approach before applying analytical tools like l'Hôpital's Rule.

• Graphing serves as a sanity check or prototype model.
• Helps to visualize the end behavior as \(x\to\infty\).
• Can reveal trends or constant values the function approaches.

Utilizing physical or software graphing tools can provide these insights and support, understanding how analytic tools like limits behave visually.