Discovering Asymptotes

6 Questions

Asymptotes help define the behavior of mathematical functions near certain values or as the input approaches infinity.

No, asymptotes typically exist for functions that have certain characteristics, such as rational functions or exponential functions.

Asymptotes are imaginary lines that a curve approaches but never touches.

Horizontal, vertical, and oblique asymptotes.

By taking the limit of the function as x approaches positive or negative infinity.

A removable vertical asymptote can be cancelled out by factoring, while a non-removable vertical asymptote cannot.

Study Notes

Asymptotes are lines that a curve approaches as the distance from the origin increases.

An asymptote is a line that a curve approaches as the input (or x-value) increases or decreases without bound, but never actually reaches.

Horizontal Asymptote: a horizontal line that a curve approaches as the input increases or decreases without bound.
Vertical Asymptote: a vertical line that a curve approaches as the input approaches a specific value.

To find the equation of a horizontal asymptote, compare the highest degree terms of the numerator and denominator of a rational function.
If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is y = a/b, where a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.
If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

A removable vertical asymptote occurs when a rational function has a factor that can be canceled out in the numerator and denominator, leaving a function with no vertical asymptote.
A non-removable vertical asymptote occurs when a rational function has a factor that cannot be canceled out in the numerator and denominator, resulting in a vertical asymptote.

6 Questions