SOLUTION: Give the equation of the horizontal asymptote of the rational function. h(x) = (7x^2-2x-4)/(9x^2-5x+2) A) None B) y = 2/5 C) y = 0 D) y = 7/9

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Question 201563: Give the equation of the horizontal asymptote of the rational function.
h(x) = (7x^2-2x-4)/(9x^2-5x+2)
A) None
B) y = 2/5
C) y = 0
D) y = 7/9
Found 2 solutions by solver91311, jsmallt9:
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!


If the degree of the denominator expression is equal to the degree of the numerator expression then there is a horizontal asymptote at where is the lead coefficient on the numerator expression and is the lead coefficient on the denominator expression.

John
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
Graphs, if they have horizontal asymptotes, will be asymptotic (i.e. the graph will get closer and closer to the asymptote) for very large positive and very large negative values of x.

The easiest way to determine what happens to the graph of a rational function like yours is to:
  1. Find the highest degree term (the term with the highest exponent) in either the numerator or the denominator
  2. Divide both the numerator and denomintor by this highest power of x (Do not include the coefficient in the division

In your function the highest power of x is . So we divide the numerator and denominator by (not or . This gives us:

Now we can figure out the horizontal asymptote, with a little basic understanding of fractions. What happens to fractions when their denominators get larger and larger? Answer: They get smaller and smaller. If the denominators get large enough the value of fraction will approach zero.

Looking at our modified version of h(x) we see a lot of fractions with x in the denominator. As x gets very large, positively or negatively, these fractions will approach zero in value. They will become negligible. What is left of h(x) if we ignore all the fractions with x in the denominator? Answer: .

So the horizontal asymptote is


In this problem the highest power of x was found in both the numerator and the denominator. If instead the highest power of x is found only in the denominator, then y = 0 will be the horizontal asymptote. (Think about this. Make up an example. It should should become clear why this is so.)

If the highest power of x is found only in the numerator, then there is no horizontal asymptote. (Again think about this and make up an example. You should see that the entire denominator will approach zero as x gets very large. When this happens the value of the whole fraction approaches infinity.).
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