Problem 28 - Thomas Calculus

Question

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of \(x :\) Divide numerator and denominator by the highest power of \(x\) in the denominator and proceed from there. Find the limits in Exercises \(23-36\). $$\lim _{x \rightarrow \infty} \frac{2+\sqrt{x}}{2-\sqrt{x}}$$

Step-by-Step Solution

Answer

The limit is -1.

Step 1: Identify the Highest Power of x in the Denominator

The denominator of the rational function is \(2 - \sqrt{x}\). The highest power of \(x\) in this expression is \(x^{1/2}\) since \(\sqrt{x} = x^{1/2}\).

Step 2: Divide Numerator and Denominator by the Highest Power of x

To simplify the expression, divide both the numerator \(2 + \sqrt{x}\) and the denominator \(2 - \sqrt{x}\) by \(x^{1/2}\).\[\lim _{x \rightarrow \infty} \frac{\frac{2}{x^{1/2}} + \frac{\sqrt{x}}{x^{1/2}}}{\frac{2}{x^{1/2}} - \frac{\sqrt{x}}{x^{1/2}}} = \lim _{x \rightarrow \infty} \frac{\frac{2}{x^{1/2}} + 1}{\frac{2}{x^{1/2}} - 1}\]

Step 3: Simplify and Evaluate the Limit

As \(x\) approaches infinity, the term \(\frac{2}{x^{1/2}}\) approaches 0, and you can simplify the expression as:\[\lim _{x \rightarrow \infty} \frac{0 + 1}{0 - 1} = \frac{1}{-1} = -1\]

Step 4: Conclusion

The limit of the function \(\lim _{x \rightarrow \infty} \frac{2+\sqrt{x}}{2-\sqrt{x}}\) is \(-1\).

Key Concepts

Division by Highest Power of x

When dealing with limits of rational functions as they approach infinity, one effective strategy is "division by the highest power of x." This technique simplifies the function, making it easier to determine the limit.
To start, identify the highest power of x in the denominator. In our example, the denominator is \(2 - \sqrt{x}\) or \(2 - x^{1/2}\). The highest power here is \(x^{1/2}\), or the square root of x.
After identifying this power, divide every term in both the numerator and the denominator by \(x^{1/2}\). This results in all terms being expressed relative to this highest value, which clears up potential infinite or undefined values.

• Numerator: \(2 + \sqrt{x}\) becomes \(\frac{2}{x^{1/2}} + 1\)
• Denominator: \(2 - \sqrt{x}\) becomes \(\frac{2}{x^{1/2}} - 1\)

This maneuver sets the stage for simplification and makes the limit more straightforward to compute.

Simplification of Limits

Once division by the highest power of x is complete, simplification becomes much more direct. The goal of simplification is to remove terms that approach zero as x approaches infinity.
Consider the expression:

• \(\lim _{x \rightarrow \infty} \frac{\frac{2}{x^{1/2}} + 1}{\frac{2}{x^{1/2}} - 1}\)

As \(x\) grows larger, the term \(\frac{2}{x^{1/2}}\) diminishes and approaches zero. This observation drastically reduces the complexity:

• The numerator \(\frac{2}{x^{1/2}} + 1\) simplifies to 1 as \(x\to\infty\).
• Similarly, the denominator \(\frac{2}{x^{1/2}} - 1\) simplifies to \(-1\) as \(x\to\infty\).

Thus, the problem reduces to evaluating a much simpler fraction \(\frac{1}{-1}\), resulting in the limit \(-1\). Keep an eye out for these kinds of simplifications, as they will often transform a seemingly complex problem into a straightforward one.

Limiting Behavior at Infinity

The limiting behavior at infinity reflects how functions behave as x becomes very large or very small. This behavior helps us understand the long-term trends of rational functions.
In the given example, after simplification, the rational function's limit is \(-1\). Understanding why the limit is reached involves observing how different terms react as x approaches infinity:

• Terms that contain x in a negative exponent (like \(\frac{2}{x^{1/2}}\)) shrink towards zero. As x enlarges, these terms contribute less to the value of the main function.
• Dominant terms, which don't disappear to zero, mainly determine the end behavior of the rational function. In this scenario, those terms in the simplified expression are \(1\) in the numerator and \(-1\) in the denominator.

By focusing on how these remaining parts influence the function, students gain insight into the stability and behavior of rational expressions at extreme values of x. This understanding is key to mastering limits and can be generalized to approach various calculus problems.