Which of the following would have a horizontal asymptote of y = b/a? Which one has a horizontal asymptote of y = 0?
( bx^3 - x^2 + 3 ) / ( ax^2 - 2 )
( bx^3 - x^2 + 3 ) / ( ax^3 - 2 )
( bx^3 - x^2 + 3 ) / ( ax^4 - 2 )
( bx^2 / a ) - 6
Please explain! :)
b/a is the same as bx^n/ax^n, so that would be
( bx^3 - x^2 + 3 ) / ( ax^3 - 2 )
as x gets large, the lower powers become insignificant, so you just have to worry about the highest power of top and bottom.
If the top has lower power than bottom, the asymptote is always y=0. That would be the 3rd function
If the top has a power one more than the bottom, there will be a slant asymptote. The function is basically bx/a + c
Okay, I think I got it... So then, of these, the limit as x approaches infinity does not exist for the second one?
ummm. The second one is the answer to the question posed. The limit is b/a.
The limit does not exist when the top power is greater than the bottom power. That would be the first one. The limit is bx/a, which grows without limit as x grows. The 4th one also grows unbounded.
To determine which of the given functions have a horizontal asymptote of y = b/a and which have a horizontal asymptote of y = 0, we need to analyze the behavior of the functions as x approaches positive or negative infinity.
First, let's review the definition of a horizontal asymptote. A horizontal asymptote is a horizontal line that a function approaches as x approaches positive or negative infinity. If the function approaches a specific value, say b/a, then the horizontal asymptote is y = b/a. If the function approaches zero, then the horizontal asymptote is y = 0.
Let's analyze each of the given functions one by one to find their horizontal asymptotes:
1. (bx^3 - x^2 + 3) / (ax^2 - 2)
In this function, the highest power of x in the numerator is x^3, and the highest power of x in the denominator is x^2. Since the degree of the numerator (3) is greater than the degree of the denominator (2), the function does not have a horizontal asymptote. Instead, it might have an oblique asymptote (a slanted line), which we can find by performing long division or synthetic division. However, since the question specifically asks for a horizontal asymptote, we can conclude that this function does not have y = b/a or y = 0 as a horizontal asymptote.
2. (bx^3 - x^2 + 3) / (ax^3 - 2)
In this function, both the numerator and denominator have the same highest power of x, which is x^3. To find the horizontal asymptote, we compare the leading coefficients of the numerator and denominator. If the leading coefficients are equal, the horizontal asymptote is y = b/a. In this case, the leading coefficient of the numerator is b, and the leading coefficient of the denominator is a. Therefore, the horizontal asymptote of this function is y = b/a.
3. (bx^3 - x^2 + 3) / (ax^4 - 2)
Similar to the previous function, both the numerator and denominator have the same highest power of x, which is x^3. However, in this case, the denominator has a higher degree (4) than the numerator (3). Whenever the denominator has a higher degree than the numerator, the function will have a horizontal asymptote of y = 0. Therefore, in this case, the horizontal asymptote is y = 0.
4. (bx^2 / a) - 6
In this function, the highest power of x is x^2 in the numerator, and there is no x variable in the denominator. Since both the numerator and denominator have a degree of 2, we compare the leading coefficients to find the horizontal asymptote. The leading coefficient of the numerator is b, and the leading coefficient of the denominator (which is a constant 1) is 1. Therefore, the horizontal asymptote of this function is y = b/a.
In summary:
- The function (bx^3 - x^2 + 3) / (ax^2 - 2) does not have a horizontal asymptote of y = b/a or y = 0.
- The function (bx^3 - x^2 + 3) / (ax^3 - 2) has a horizontal asymptote of y = b/a.
- The function (bx^3 - x^2 + 3) / (ax^4 - 2) has a horizontal asymptote of y = 0.
- The function (bx^2 / a) - 6 has a horizontal asymptote of y = b/a.