Find Horizontal Asymptote of Rational Function F(x)

In summary, for the function f(x) = (6x^2-17x-3)/(3x+2), there are no horizontal asymptotes, but there is a vertical asymptote at x = -2/3. There may also be a diagonal asymptote, but this depends on whether or not it has been taught.
  • #1
shorty888
6
0
F(x)= 6x^2-17x-3/3x+2, find horizontal asymptote
 
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  • #2
shorty888 said:
F(x)= 6x^2-17x-3/3x+2, find horizontal asymptote

Do you mean $f(x) = \frac{6x^2-17x-3}{3x+2}$

A horizontal asymptote is a line where a function approaches but doesn't quite reach there. An example is y=0 for f(x) = e^x

There are no horizontal asymptotes in your case, there is a vertical one where 3x+2 = 0 but no horizontal ones.

nb: Please take care with brackets, I've guessed at what you meant but that isn't the literal meaning of what you wrote which would be $F(x) = 6x^2-17x-x+2$
 
  • #3
Yes I mean fraction

shorty888 said:
F(x)= 6x^2-17x-3/3x+2, find horizontal asymptote
 
  • #4
SuperSonic4 said:
Do you mean $f(x) = \frac{6x^2-17x-3}{3x+2}$

A horizontal asymptote is a line where a function approaches but doesn't quite reach there. An example is y=0 for f(x) = e^x

There are no horizontal asymptotes in your case, there is a vertical one where 3x+2 = 0 but no horizontal ones.

nb: Please take care with brackets, I've guessed at what you meant but that isn't the literal meaning of what you wrote which would be $F(x) = 6x^2-17x-x+2$
There is also a diagonal asymptote but a word from the OP will be required to find out if such asymptotes have been taught.
 
  • #5


The horizontal asymptote of a rational function is the value that the function approaches as x increases or decreases without bound. To find the horizontal asymptote of the given function F(x) = (6x^2-17x-3)/(3x+2), we can use the following steps:

1. Simplify the rational function by dividing the numerator and denominator by the highest power of x. In this case, the highest power of x is 2, so we can divide both the numerator and denominator by x^2.

F(x) = (6x^2-17x-3)/(3x+2) = (6-17/x-3/x^2)/(3/x+2/x^2)
= (6/x^2-17/x-3)/(3/x^2+2/x)

2. As x approaches infinity or negative infinity, the terms with the highest powers of x will dominate the fraction. In this case, as x approaches infinity, the fractions 6/x^2 and 3/x^2 will approach 0, while the fractions -17/x and 2/x will approach 0. Therefore, the simplified function becomes:

F(x) = 0/0

3. According to the rules of limits, when a fraction has a numerator and denominator both approaching 0, the limit of the fraction can be found by dividing the coefficients of the highest power terms. In this case, the coefficient of x^2 in the numerator is 6, and the coefficient of x^2 in the denominator is 3. Therefore, the limit is:

lim(x->∞) F(x) = 6/3 = 2

4. This means that as x approaches infinity, the function F(x) will approach the value of 2. Therefore, the horizontal asymptote of the function is y=2.

In conclusion, the horizontal asymptote of the given rational function F(x) = (6x^2-17x-3)/(3x+2) is y=2. This means that as x increases or decreases without bound, the function will approach the value of 2.
 

FAQ: Find Horizontal Asymptote of Rational Function F(x)

What is a horizontal asymptote?

A horizontal asymptote is a straight line that a function approaches as the input values get larger and larger or smaller and smaller. It is represented by the equation y = c, where c is a constant.

How do you find the horizontal asymptote of a rational function?

To find the horizontal asymptote of a rational function, you need to compare the degrees of the numerator and denominator. If the degree of the numerator is smaller than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator. If the degree of the numerator is larger than the degree of the denominator, there is no horizontal asymptote.

Can a rational function have more than one horizontal asymptote?

No, a rational function can only have at most one horizontal asymptote. This is because the behavior of a rational function as the input values get larger or smaller is determined by the degrees of the numerator and denominator.

What if the degrees of the numerator and denominator are the same?

If the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator. For example, if the rational function is f(x) = (2x + 3)/(4x + 5), the horizontal asymptote is y = 2/4 or y = 1/2.

Can the horizontal asymptote of a rational function ever intersect the graph of the function?

No, the horizontal asymptote of a rational function never intersects the graph of the function. This is because, by definition, the horizontal asymptote is a line that the function approaches but never touches.

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