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Chapter 3: Problem 38

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

Short Answer

Expert verified
The horizontal asymptote of the function \(f(x)=\frac{15x}{3x^{2}+1}\) is y=0.

Step by step solution

01

Identify the type of function

In the function \(f(x)=\frac{15x}{3x^{2}+1}\), the numerator is a polynomial of degree 1 and the denominator is a polynomial of degree 2. This is a rational function.
02

Compare the degree of numerator and denominator

The degree of the numerator (1) is less than the degree of the denominator (2).
03

Find the horizontal asymptote

For a rational function where the degree of the numerator is less than the degree of the denominator, the value of the function approaches zero as x approaches infinity or negative infinity. Hence, the horizontal asymptote is y=0.

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Most popular questions from this chapter

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$ \frac{x}{2 x+6}-\frac{9}{x^{2}-9} $$

Use a graphing utility to graph \(y=\frac{1}{x^{\prime}}, y=\frac{1}{x^{3}},\) and \(\frac{1}{x^{5}}\) in the same viewing rectangle. For odd values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

Can the graph of a polynomial function have no y@intercept? Explain.

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the ZOOM OUT feature to show that f and g have identical end behavior. \(f(x)=x^{3}-6 x+1, g(x)=x^{3}\)

Use long division to rewrite the equation for \(g\) in the form $$ \text {quotient}+\frac{\text {remainder}}{\text {divisor}} $$ Then use this form of the function's equation and transformations \( \text { of } f(x)=\frac{1}{x} \text { to graph } g \). $$ g(x)=\frac{3 x-7}{x-2} $$
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