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Chapter 4: Problem 55

Use a graphing utility to determine whether or not the graph of \(f\) has a horizontal asymptote. Confirm your findings analytically. $$f(x)=\sqrt{x^{2}+2 x}-x$$

Short Answer

Expert verified
The x-axis (y = 0) is a horizontal asymptote of the given function.

Step by step solution

01

Graph the function

To graph the function, use a graphing utility such as Desmos, GeoGebra, or a graphing calculator. Enter the function \(f(x) = \sqrt{x^{2}+2x} - x\) and observe the graph. Note the behavior of the function as x approaches positive and negative infinity to determine if there's a line that the graph is approaching.
02

Inspect the graph visually

Looking at the graph, it appears that as x approaches infinity or negative infinity, the graph of the function seems to be getting closer and closer to the x-axis. Hence it can be inferred that x-axis (y=0) is the horizontal asymptote of the graph.
03

Confirm analytically

To confirm the presence of a horizontal asymptote analytically, we need to evaluate the limit of the function as x approaches positive and negative infinity. In this case, due to the structure of the function, it would be beneficial to use the property \(a^2 - b^2 = (a-b)(a+b)\) to rewrite the function as \(f(x) = (x^{2} - x^{2} + 2x) / (\sqrt{x^{2} + 2x} + x)\). Simplifying this yields \(f(x) = 2x / (\sqrt{x^{2} + 2x} + x)\). Now calculate the limit as x approaches infinity and negative infinity. If the limit produces a real number, that is the value of your horizontal asymptote.
04

Evaluate Limits

Evaluate \(\lim_{{x\to\infty}} (2x/(\sqrt{x^{2}+2x} + x)) = 0\) and \(\lim_{{x\to-\infty}} (2x/(\sqrt{x^{2}+2x} + x)) = 0\). As the limits as x approaches both positive and negative infinity is zero, this confirms that the x-axis (y = 0) is a horizontal asymptote of the graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Utility
To analyze a function visually, a graphing utility is incredibly useful. Whether you鈥檙e using Desmos, GeoGebra, or a standard graphing calculator, these tools can help quickly visualize the behavior of the function.
Simply input your function, such as the given function \( f(x) = \sqrt{x^{2}+2x} - x \), and you will see its graphical representation.
  • Look for points where the function levels off or approaches a straight line as it moves towards 卤鈭� (positive or negative infinity).
  • Notice any horizontal lines that the graph seems to approach but not touch 鈥� these could be horizontal asymptotes.
The graphing utility provides a quick visual affirmation, but always remember to verify your results through additional methods to ensure accuracy.
Horizontal Asymptote
In calculus, a horizontal asymptote is a y-value that a function approaches but never quite reaches as x tends to 卤鈭�. It gives you a sense of the end behavior of the function.
For our function \( f(x) = \sqrt{x^{2}+2x} - x \), you observe that as \( x \) goes towards infinity or negative infinity, the graph appears to approach the x-axis (\( y=0 \)). This suggests that \( y=0 \) might be a horizontal asymptote.
  • The horizontal asymptote can often be deduced by visually examining the graph.
  • However, for a thorough analysis, it鈥檚 important to confirm findings analytically, particularly via limits.
Understanding horizontal asymptotes allows you to better predict the behavior of functions at extreme values of x, which is valuable for graphing and real-world applications.
Analytic Methods
To confirm the presence of a horizontal asymptote analytically, especially with the function \( f(x) = \sqrt{x^{2}+2x} - x \), you need to evaluate limits. When you explore limits, you're determining the value that a function approaches as \( x \) gets very large or very small.
Use algebraic manipulation to simplify first:
  • Express \( f(x) \) as \( \frac{2x}{\sqrt{x^2+2x}+x} \)
Analyzing this limit:
  • As \( x \to \infty \), you evaluate \( \lim_{{x\to\infty}} \frac{2x}{\sqrt{x^2+2x}+x} = 0 \).
  • Similarly, as \( x \to -\infty \), \( \lim_{{x\to-\infty}} \frac{2x}{\sqrt{x^2+2x}+x} = 0 \).
Both limits confirm that \( y=0 \) is indeed a horizontal asymptote.
Analytical methods reinforce what you might initially see on a graph and provide a more rigorous mathematical validation.

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Most popular questions from this chapter

Sketch the graph of the function showing all vertical and oblique asymptotes. $$f(x)=\frac{1+x-3 x^{2}}{x}$$

An object that weighs 150 pounds on the surface of the earth will weigh \(150\left(1+\frac{1}{4000} r\right)^{-2}\) pounds when it is \(r\) miles above the earth. Given that the altitude of the object is increasing at the rate of 10 miles per minute, how fast is the weight decreasing which the object is 400 miles above the surface?

A tour boat heads out on a 100 -kilometre sight-seeing trip. Given that the fund costs of operating the boat total 2500 dollar per hour, that the cost of fuel varies directly with the square of the speed of the boat, and at 10 kilometres per hour the cost of the fuel is 400 dollar per hour, find the speed test minimizes the boat owner's expenses. Is the speed that minimizes the owner's expenses dependent on the length of the trip?

Set \(f(x)=\frac{x^{3}-x^{1 / 3}}{x}\). Show that \(f(x)-x^{2} \rightarrow 0\) as \(x \rightarrow \pm \infty .\) This says that the graph of \(f\) is asymptotic to the parabola \(y=x^{2} .\) Sketch the graph of \(f\) and feature this asymptotic behavior.

Sketch the graph of the function using the approach presented in this section. $$f(x)=\frac{x^{2}}{\sqrt{x^{2}-2}}$$
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