How do you find the vertical, horizontal or slant asymptotes for #(4x^2+5)/( x^2-1)#?
Ok, let's start with the vertical asymptotes. You know that a function is not defined when a denominator equals
This gives us:
I don't know how advanced your problems will get, but there is now a possibility of either a vertical asymptote or a removable discontinuity. If you have no idea what a removable discontinuity is, you can probably skip this next part.
In order to know if you have a removable discontinuity, try factoring both the top and the bottom. If any of the factors cancel, then you have a removable discontinuity at that point. Take for example:
This factors into:
Now you notice that the
This makes a line, which is pretty easy to graph. However, because
However, the top part of your problem doesn't even factor, so there cannot be a removable discontinuity giving you vertical asymptotes at
End of Removable Discontinuities
Now on to the horizontal and oblique (or slant as you called it) asymptotes.
This is done by doing the following:
First take the highest powered variables on the top and the bottom and remove the rest. In your case this would be
We can do this because horizontal and oblique asymptotes are when
Now, determine which power is greater;
If the top is bigger:
Simplify! What you end up with is your oblique asymptote
If the bottom is bigger:
You got off easy. The horizontal asymptote is just
If they are equal (like your problem):
Divide the coefficients (in your case it's
That means that your horizontal asymptote is
Hope this helped!
Jonathan 'JMoney' Moore