Give an example of a rational function that has no horizontal asymptote and a vertical asymptote at
x = 1.
how about
y = x^3/(x-1)
confirmed:
http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E3%2F%28x-1%29
To find a rational function that has no horizontal asymptote and a vertical asymptote at x = 1, we can start by using the standard form of a rational function:
f(x) = (P(x))/(Q(x))
where P(x) and Q(x) are polynomials.
To have a vertical asymptote at x = 1, we need a factor of (x - 1) in the denominator, Q(x). This means that Q(x) cannot be equal to zero when x = 1, which guarantees the vertical asymptote.
To have no horizontal asymptote, we need P(x) to have a higher degree than Q(x). This means that the highest degree terms of P(x) and Q(x) will not cancel each other out as x goes to infinity or negative infinity.
An example of such a rational function is:
f(x) = (x^2 + 1) / (x - 1)
In this example, the degree of the polynomial P(x) (x^2 + 1) is 2, which is higher than the degree of the polynomial Q(x) (x - 1), which is 1. Therefore, there will be no horizontal asymptote.
And since Q(x) is equal to zero when x = 1, we have a vertical asymptote at x = 1.
To find a rational function that has no horizontal asymptote and a vertical asymptote at x=1, we can start by considering the general form of a rational function:
f(x) = (ax^n + bx^(n-1) + ... + k) / (cx^m + dx^(m-1) + ... +w)
To have a vertical asymptote at x=1, we need a factor of (x-1) in the denominator, since the denominator should not equal zero in order to avoid dividing by zero. Therefore, let's set:
c(x - 1) = cx - c
Now, let's consider the numerator. Since we want to find a rational function with no horizontal asymptote, the degree of the numerator (highest power of x) should be greater than the degree of the denominator. This means that n > m.
So, let's say n = m + 1.
With these considerations, our rational function becomes:
f(x) = (ax^(m + 1) + bx^m + ... + k) / (cx - c)
Here, a, b, c, k, and w are constants that can be any real numbers. We can adjust these constants to fit specific criteria or create specific graphs. For now, we'll keep them as arbitrary constants.
This rational function will have a vertical asymptote at x=1 since the denominator becomes zero at that point, and the numerator will only approach zero as x approaches infinity or negative infinity. However, because the numerator has a higher degree than the denominator, the function does not have a horizontal asymptote. The function will exhibit slant asymptotes or oblique asymptotes instead.
It's worth noting that multiple rational functions can fulfill the given conditions. The example given above is just one possible solution. You can experiment with different values of the constants to create other rational functions with the desired properties.