Respuesta :
Vertical asymptote:
Find the restriction on x. This is the easiest of the three asymptotes you will need to find (even if only two can show at a time). As a hyperbola is in the form: [tex]\frac{1}{x}[/tex], you only need to find the restriction on the denominator, namely denominator can never be zero. Hence, let the denominator equal to zero to find the vertical asymptote.
Horizontal asymptote:
Find the restriction on y. To do this, you need to simplify the top and bottom to its lowest terms. If it simplifies to a form such as: [tex]a + \frac{b}{x}[/tex], then the horizontal asymptote becomes y = a. You need to think to yourself, as x grows to infinite, and shrinks to negative infinite, what happens to the function? Does it slowly curve to a stop?
Oblique asymptote:
This is a pretty rare kind, but it still exists, so don't be naive to this sort of asymptote. This is a form of horizontal and vertical asymptote, only it's at an angle. That is, this asymptote is a set of x and y-coordinates that work in unison to produce a curvature or line.
Let's consider: [tex]f(x) = \frac{x^{2} - 6x + 7}{x + 5}[/tex]
Now, in normal term, a horizontal asymptote would have a degree higher in the denominator than in the numerator. However, it's flipped in this case.
Now, you will need to long divide this set of polynomials to yield a straight y = x line, except it's been moved 11 units to the right to yield a y = x - 11 line.
Remember: these exist because the highest power is in the numerator and not the denominator.
Find the restriction on x. This is the easiest of the three asymptotes you will need to find (even if only two can show at a time). As a hyperbola is in the form: [tex]\frac{1}{x}[/tex], you only need to find the restriction on the denominator, namely denominator can never be zero. Hence, let the denominator equal to zero to find the vertical asymptote.
Horizontal asymptote:
Find the restriction on y. To do this, you need to simplify the top and bottom to its lowest terms. If it simplifies to a form such as: [tex]a + \frac{b}{x}[/tex], then the horizontal asymptote becomes y = a. You need to think to yourself, as x grows to infinite, and shrinks to negative infinite, what happens to the function? Does it slowly curve to a stop?
Oblique asymptote:
This is a pretty rare kind, but it still exists, so don't be naive to this sort of asymptote. This is a form of horizontal and vertical asymptote, only it's at an angle. That is, this asymptote is a set of x and y-coordinates that work in unison to produce a curvature or line.
Let's consider: [tex]f(x) = \frac{x^{2} - 6x + 7}{x + 5}[/tex]
Now, in normal term, a horizontal asymptote would have a degree higher in the denominator than in the numerator. However, it's flipped in this case.
Now, you will need to long divide this set of polynomials to yield a straight y = x line, except it's been moved 11 units to the right to yield a y = x - 11 line.
Remember: these exist because the highest power is in the numerator and not the denominator.
