Problem 46 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(45-48\) , find the limit. (Hint: Treat the expression as a fraction whose denominator is 1 , and rationalize the numerator.) Use a graphing utility to verify your result. $$ \lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}+x}\right) $$

Step-by-Step Solution

Answer

The limit as \(x\) approaches infinity of the function \(x-\sqrt{x^{2}+x}\) is -1.

Step 1: Rationalize the numerator

To get rid of the radical, multiply both the numerator and denominator of the fraction by the conjugate of the numerator, which is \(\left(x+\sqrt{x^{2}+x}\right)\). So, we get: \[ \lim _{x \rightarrow \infty}\frac{(x-\sqrt{x^{2}+x})(x+\sqrt{x^{2}+x})} {1*(x+\sqrt{x^{2}+x})} \]

Step 2: Simplify the expression

Multiply through to simplify: \[ = \lim _{x \rightarrow \infty}\frac{x^2-(x^2+x)} {x+\sqrt{x^{2}+x}} \] \ = \[ \lim _{x \rightarrow \infty}\frac{-x} {x+\sqrt{x^{2}+x}} \]

Step 3: Substitute \(x\) by infinity

Substitute \(x\) by infinity: \[ = \frac{-\infty} {\infty+\sqrt{\infty^{2}+\infty}} \] Since infinity plus or minus infinity is still infinity, we have a \(\frac{-\infty}{\infty}\) situation, which simplifies to -1. So: \[ = -1 \]

Key Concepts

Rationalize the numerator

When you encounter expressions that include radicals, it can be helpful to rationalize the numerator. Rationalization is a technique used to eliminate the radical, which simplifies the expression and makes solving limits easier. In this context, our target expression is \(x - \sqrt{x^2 + x}\). To rationalize the numerator:

• Multiply the numerator and the denominator by the conjugate of the numerator.
• The conjugate is \(x + \sqrt{x^2 + x}\). Conjugates help by creating a difference of squares, eliminating the radical term efficiently.
• Computing \((x - \sqrt{x^2 + x})(x + \sqrt{x^2 + x})\) yields \(x^2 - (x^2 + x)\), simplifying to \(-x\).

Rationalizing not only makes the expression easier to handle but also prepares it for further analysis, such as taking the limit.

Graphical Verification

Graphical verification is a handy way to visually confirm the behavior of functions as they approach a limit. Here's how it works:

• Use a graphing utility, such as a calculator or software like Desmos or GeoGebra, to plot the function \(f(x) = x - \sqrt{x^2 + x}\).
• Examine the graph as \(x\) approaches infinity. Notice how the function stabilizes as its value trends closer to -1.
• The graphical plot will show the horizontal asymptote, delineating towards which value the function settles as \(x\) infinitely increases.

This graphical approach offers an intuitive understanding of limits and serves as a cross-check to analytical solutions.

Infinite Limits

Infinite limits refer to the behavior of a function as the input value grows indefinitely large. As \(x\) approaches infinity in our equation, we explore the behavior of the function \(x - \sqrt{x^2 + x}\):

• Infinite limits often create expressions like \(\frac{-\infty}{\infty}\). While this might seem undefined at first, careful analysis can demonstrate a distinct limit.
• In simplifying \(\frac{-x}{x + \sqrt{x^2 + x}}\), note that as \(x\) tends to infinity, both \(x\) and \(\sqrt{x^2 + x}\) approximate to \(x\), making their ratio close to 1.
• Thus, the entire expression simplifies down to \(-1\).

Understanding infinite limits helps you grasp asymptotic behavior, offering key insights into the function's behavior at numerical extremes.