Asymptopes

An asymptote is a line that a curve approaches (but never actually touches), as it heads towards infinity

Three types: horizontal, vertical, and oblique.

The degree of the polynomials will lead us to what type of asymptope we have.

Degree of the numerator is less than the denominator, there will be a horizontal asymptope. (specifically, it will be at y=0). For example:

Degree of the numerator is equal to the denominator, there will be a horizontal asymptope. (specifically, it will be at the ratio of the coefficients of the highest degree terms.) For example:

 $f\left(x\right)=\frac{9x^2+5}{3x^2+3}$  will have a horizontal asymptope at y=3.

There will never be a horizontal asymptope when the degree of the numerator is greater then the denominator.

Range: all real numbers.

Focus on the denominator!

Whenever the denominator is 0 (the function is undefined), there will be a vertical asymptope.

Finding the domain is vital for vertical asymptopes.

For example  $f\left(x\right)=\frac{1}{x}$  will have a vertical asymptope at x=0.