1. ## Horizontal Asymptote Defied

I don't know whether to post this in an Algebra section or a Calculus section so I just chose Algebra. I have a small understanding of limits if it is possible to solve the following equation in that manner. To the point, the following equation should have a horizontal asymptote at 1 and a vertical asymptote at -2 and 1, all of these are true except at the point (-3.5, 1) exists and there is no possibility of it being a hole because there is no canceling factor in the numerator and the denominator for two reasons:
1) The numerator cannot be factored
2) Both of the factors of the denominator are vertical asymptotes
If someone could attempt to rationalize this for me I would be very much appreciative. I love math and spend countless hours working with it, for fun. I was studying Pre-Calculus in my free time in my Algebra 2 w/ Trig. class, I had a lot of free time due to my extensive understanding of mathematics.

Here is the original equation I am referring to:
(x^2 + 3x + 1)/(x^2 + x - 2)

2. I'm not sure what's troubling you about this. You have correctly identified the asymptotes it seems, but plugging -3.5 into that expression does not give 1, though if it did I don't see why that would be an issue.

Incidentally, the numerator does factor (apply quadratic formula, etc).

3. Pardon me, I mean that the point that I don't understand is at (-1.5, 1). Shouldn't this point not exist because there is a horizontal asymptote at 1? Therefore, the line should never cross/touch 1.

4. Originally Posted by sudox
Pardon me, I mean that the point that I don't understand is at (-1.5, 1). Shouldn't this point not exist because there is a horizontal asymptote at 1? Therefore, the line should never cross/touch 1.
You have the misconception that a graph cannot touch or cross a horizontal asymptote. Such a misconception is typically the result of either lazy teaching or incompetence (or both) at the lower levels (Teacher: "A graph can never cross an asymptote.") The fact that the curriculum at lower levels invariably includes only examples of graphs that do not cross their horizontal asymptotes (such as y = 1/x and y = 1/x^2) serves only to re-inforce this unfortunate misconception. Better learning would occur if the curriculum included examples such as y = x/(x^2 + 1).

Now, listen very carefully:

A graph can never touch or cross a vertical asymptote. But it IS possible for a graph to touch or cross a horizontal asymptote. Go back and look very carefully at the definition of each type of asymptote to understand why.

The classic example of a graph crossing its horizontal asymptote is the graph of the function y = x/(x^2 + 1). The x-axis is a horizontal asymptote but the graph obviously passes through the origin.

5. Thank you for revealing that information to me, it is truly pathetic that a mathematics teacher doesn't know this and even more pathetic that it isn't taught in this manner. I looked up Horizontal Asymptotes and got a confirmation of your explanation here:
Horizontal Asymptotes