Asymptotes

If a function f gets close to a certain number L when x gets larger and larger, then we say that the limit as x goes to infinity is L and we write: . Likewise, if f gets close to L when x gets smaller and smaller, then the limit as x goes to negative infinity is L and we write: . In both cases, the line y = L is a horizontal asymptote of f.

Example: because when x is very large, is close to 0. The x -axis is a horizontal asymptote of the function :

If a function gets larger and larger as x gets close to a number a, then it “goes to infinity” and we write: . The line x = a is a vertical asymptote of f. Similarly, if gets smaller and smaller as x gets close to a, then it “goes to negative infinity” and we write: . Again, the line x = a is a vertical asymptote of f.

An important result:

If then . This is because one over a very large positive number and one over a huge negative number are both close to 0.

Inclined asymptotes have an equation y = kx + q and their position is arbitrary except vertical. In order a straight line y = kx + q be an asymptote, the coefficients k and q must satisfy at least one pair of the following conditions

and (k and b are numbers).

Naturally these limits must be finite real numbers. A certain function can have maximally two inclined asymptotes.