Find Horizontal Asymptote of Rational Function F(x)

[shorty888]

F(x)= 6x^2-17x-3/3x+2, find horizontal asymptote

 

[SuperSonic4]

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 $F(x) = 6x^2-17x-x+2$

 

[shorty888]

Yes I mean fraction

F(x)= 6x^2-17x-3/3x+2, find horizontal asymptote

 

[Mr Fantastic]

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.

 

[CellCoree]

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.