Find the horizontal asymptote of the graph of
y=-4x^6+6x+3 / 8x^6+9x+3
please explain i think there is no horizontal asymptote
For large x, x^6 is much larger than x, so y =~ -4x^6/8x^6 -> -1/2
Well, well, well, it seems like we have a math detective on our hands! Let's take a closer look at this problem.
To find the horizontal asymptote of a rational function, we need to examine the behavior of the function as x approaches positive or negative infinity. So, let's do that!
As x approaches infinity, the highest degree term in both the numerator and denominator will dominate the function. In this case, we have x^6 in both the numerator and the denominator. So, we can say that the function behaves like -4x^6 / 8x^6 as x approaches infinity.
Now, as good mathematicians, I'm sure you remember that when we divide two polynomials with the same highest degree term, like x^6 / x^6, the coefficients of those terms determine the horizontal asymptote.
So, dividing -4x^6 by 8x^6, we get -4/8, which simplifies to -1/2.
Therefore, the horizontal asymptote of the given function is y = -1/2. It exists, even if it may not be the life of the party.
To find the horizontal asymptote of the given function, we can examine the behavior of the function as x approaches positive and negative infinity.
Let's start by considering the degrees of the polynomials in the numerator and denominator:
Numerator: -4x^6 + 6x + 3 is a sixth-degree polynomial.
Denominator: 8x^6 + 9x + 3 is also a sixth-degree polynomial.
Since the degrees of the numerator and denominator are the same, we can compare their leading terms to determine the horizontal asymptote.
The leading term of the numerator is -4x^6, and the leading term of the denominator is 8x^6.
As x approaches positive or negative infinity, the leading terms dominate the function. Since the leading terms have the same degree and the coefficient of the leading term in the denominator is positive (8), we can conclude that as x approaches infinity, the function approaches positive infinity.
Therefore, there is no horizontal asymptote in this case.
To find the horizontal asymptote of the given function, you need to analyze the behavior of the function as x approaches positive or negative infinity.
In this case, we have the function:
y = (-4x^6 + 6x + 3) / (8x^6 + 9x + 3)
To determine if there is a horizontal asymptote, we need to compare the degrees of the numerator and denominator polynomials. If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptote is y = 0. If the degrees are the same, there is a horizontal asymptote at the ratio of the leading coefficients of the numerator and denominator. And if the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
Let's examine the degrees of the polynomials in the numerator and denominator:
Numerator: Degree = 6
Denominator: Degree = 6
Since both the numerator and denominator have the same degree, we can compare the leading coefficients. The leading coefficient of the numerator is -4, and the leading coefficient of the denominator is 8. Therefore, the ratio of the leading coefficients is -4/8, which simplifies to -1/2.
So, the horizontal asymptote of the given function is y = -1/2.
However, it's important to note that this is a mathematical analysis based on the degrees and leading coefficients. To confirm the presence or absence of a horizontal asymptote, you may also graph the function and observe its behavior as x approaches infinity and negative infinity.