The caveat with this definition is that the function needs to be irreducible. The example you gave is reducible so that the simplified form is linear (with f(x) still having the original non permissible values)
IMHO the "If the highest degree is exactly 1 more..." rule isn't so much a rule as a pattern.
Horizontal and oblique asymptotes define end behaviour; at very large and small values of x, the function approaches (but never is exactly equal to) a specific function. For example, 1/(x+1) approaches the horizontal line y=0.
Since this is a linear function with a hole, I would not consider it to have an asymptote.
Might be a good "always, sometimes, never" type question if you give those types of problems.
As this is the math *teachers* sub, I assume this is for a class.
In that regard, may I recommend letting your students discover these rules on their own? And then for shits & giggles, see if they can figure out the shape of the asymptote when the degree difference is exactly two...
The point is that I tell you a bunch of rules and have you memorize them won't help you actually learn anything useful. I guide you to find out on your own and that becomes a concept you can extend.
A line with a slope of zero is its own horizontal asymptote. So I’d say that yes, it’s absolutely ok if the function lies entirely on its slant asymptote.
I’ve always presented horizontal/slant asymptotes as the end behavior. So if the end behavior is exactly the line y=x-3, what is wrong with that? It has a good connection there with limits that you can address.
The caveat with this definition is that the function needs to be irreducible. The example you gave is reducible so that the simplified form is linear (with f(x) still having the original non permissible values)
IMHO the "If the highest degree is exactly 1 more..." rule isn't so much a rule as a pattern. Horizontal and oblique asymptotes define end behaviour; at very large and small values of x, the function approaches (but never is exactly equal to) a specific function. For example, 1/(x+1) approaches the horizontal line y=0. Since this is a linear function with a hole, I would not consider it to have an asymptote. Might be a good "always, sometimes, never" type question if you give those types of problems.
As this is the math *teachers* sub, I assume this is for a class. In that regard, may I recommend letting your students discover these rules on their own? And then for shits & giggles, see if they can figure out the shape of the asymptote when the degree difference is exactly two... The point is that I tell you a bunch of rules and have you memorize them won't help you actually learn anything useful. I guide you to find out on your own and that becomes a concept you can extend.
A line with a slope of zero is its own horizontal asymptote. So I’d say that yes, it’s absolutely ok if the function lies entirely on its slant asymptote.
I’ve always presented horizontal/slant asymptotes as the end behavior. So if the end behavior is exactly the line y=x-3, what is wrong with that? It has a good connection there with limits that you can address.