I am working on a math problem for class and am stumped by the nature of its horizontal asymptote.
The equation is f(x) = x / (x+5)(x-2).
Based on the rules for horizontal asymptotes given to me in class, when the degree of the leading coefficient for the denominator is larger than the degree of the leading coefficient for the numerator, the horizontal asymptote is always Y = 0. Since the numerator is X and the denominator is X^2, the denominator is larger so Y = 0.
For the equation above, the horizontal asymptote holds true as X goes towards positive and negative infinity outside of the vertical asymptotes (X = -5 & X = 2). However, inbetween the two vertical asymptotes, the graph crosses the X axis at (0,0).
I am curious why the function behaves this way, and if there is any vocabulary for this phenomenon.
The fact that the function passes through the origin is a simple consequence of the zero at $x=0$ of the function, i.e. the numerator (but not the denominator) is zero there. This is not related to the horizontal nor to the vertical asymptotes.
When you say that the horizontal asymptote is "valid" at $\pm\infty$ but not "inside the vertical asymptotes", it suggests that you think it could be "valid" elsewhere, but that's not something horizontal asymptotes do or describe. They only tell you something about the behaviour of the function for arbitrary large ($x\to+\infty$) and arbitrary small ($x\to-\infty$) values; anything can happen "in between".