(conceptual) question about asymptotes

[ozil]

I don't think this question requires the template. Basically can some one simply explain something to me regarding asymptotes:

The rules are that for horizontal the numerator has to have a higher power. For slant / oblique the numerator has to be just 1 higher than the denominator.

How do you not confuse the two? How can you tell which is which?

 

[Let'sthink]

I think one can derive those observations by finding out dy/dx and study its behavior as x or/and y tend to infinity. I do not think there can be any other explanation than mathematical based on calculus and algebra, which includes the process of evaluating limits. Also what you are telling may be necessary but not sufficient condition. Think over that too.

 

[Mark44]

I don't think this question requires the template. Basically can some one simply explain something to me regarding asymptotes:

The rules are that for horizontal the numerator has to have a higher power.

From "numerator has to have a higher power" I assume you're talking about rational functions, which are quotients of polynomials. If the degree of the numerator (function on top) is one larger than the degree of the denominator (function on bottom), there is an oblique asymptote. (If the degree of the numerator is larger by two or more, there is no straight line asymptote.)

If the degree of the denominator is equal to the degree of the numerator, there is a horizontal asymptote that is either above or below the horizontal axis. Its equation is ##y = \frac{a_n}{b_n}##, where ##a_n## is the coefficient of the highest degree term in the numerator, and ##b_n## is the coefficient of the highest degree term in the denominator.

If the degree of the numerator is less than that of the denominator, the x-axis is the horizontal asymptote.

For slant / oblique the numerator has to be just 1 higher than the denominator.

How do you not confuse the two? How can you tell which is which?