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Chapter 3: Problem 40

Find the horizontal asymptote, if there is one, of the graph of each rational function. $$ g(x)=\frac{15 x^{2}}{3 x^{2}+1} $$

Short Answer

Expert verified
The line \(y=5\) is the horizontal asymptote for the given function.

Step by step solution

01

Analyze degrees of polynomials

Look at the degrees of the polynomials in the numerator and denominator. Both are of degree 2.
02

Apply the rule for horizontal asymptotes

If the degrees of the polynomials in the numerator and denominator are equal, then the horizontal asymptote is given by the ratio of the leading coefficients of the numerator and the denominator. Here, the leading coefficient for the numerator is 15 and for the denominator is 3.
03

Calculate the horizontal asymptote

The horizontal asymptote would be the ratio of the leading coefficients, that is, \(y = \frac{15}{3} = 5\). Therefore, the line \(y=5\) is the horizontal asymptote for \(g(x) = \frac{15x^2}{3x^2+1}\).

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Most popular questions from this chapter

Exercises 113鈥�115 will help you prepare for the material covered in the next section. Rewrite \(4-5 x-x^{2}+6 x^{3}\) in descending powers of \(x\)

In Exercises 94鈥�97, use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-2 x^{3}+6 x^{2}+3 x-1$$

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. It is possible to have a rational function whose graph has no \(y\) -intercept.

What is meant by the end behavior of a polynomial function?

In Exercises \(98-99,\) use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the \([\mathrm{ZOOMOUT}]\) feature to show that \(f\) and \(g\) have identical end behavior. $$f(x)=x^{3}-6 x+1, \quad g(x)=x^{3}$$
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