Asymptotes VS. Holes
Does anyone remember/know the exact rule for when a graph will have a vertical asymptote and when it will have a hole in it instead? Because I would think that any value for x that would make the denominator of an expression 0 would create a hole, but apparently sometimes it's an asymptote instead. I think in class we did one problem like this on Friday, but I can't seem to find understandable notes about it. Is it only a hole when you can cancel out a factor from both the numerator and the denominator? Like 4.5.41... The function is f(x)=(2x^2-x-6)/(x^3+x^2-6x). You can factor (x-2) out of the numerator and the denominator, and the book said this gives the graph a hole at x=2. But when a factor doesn't entirely disappear by canceling out, there seems to be an asymptote instead. Hm... i'm bad at explaining. If anyone understands the jist of what i'm talking about and has any ideas about it, let me know.
posted by Casey Blue at 3:21 PM
1 Comments:
I don't have an exact rule, but one way to think about it that i find helpful is like the exercise we did on the group work where we completed the chart.By plugging in values for x that are near the zero of the denominator, you can see a trend in the outputs. if the values seem to approach a certain number, then it is an asymptote. If not, it is most likely a hole.
GROUP WORK EXAMPLE:
does the graph of this limit have a hole or a vertical asymptote at x=2?
(2^x-4)/(x-2)
since 2 is the zero of the denominator, you plug in values near 2 (1.9, 1.99, 2.01, etc). since the outputs of these values are not approaching some number on either side of 2, it can be concluded that it is a hole.
Post a Comment
<< Home