2.6 Asymptotes and Intercepts

2.6 Asymptotes and Intercepts

Definition of Vertical and Horizontal Asymptotes

Vertical- The line x=a is a vertical asymptote of the graph of f if f(x) approaches infinity or f(x) approaches negative infinity as x approaches a, either from the right or left
Horizontal- The line y=b is a horizontal asymptote of the graph of f if f(x) approaches b as x approaches infinity or x approaches negative infinity.  
                                                         

               

The function above has 2 vertical asymptotes and one horizontal asymptote

vertical: x=1 and x=-1

horizontal: y=1

Rational Function example:

The vertical asymptotes come from the zeroes of the denominator

• Two vertical asymptotes are at +-3/2
• Since the degrees of the numerator and denominator are the same then this rational has a non zero horizontal asymptote
• The horizontal asymptote is found by dividing the leading terms

• Horizontal asymptote y=1/4

Example 2:

• y=2/3 is the horizontal asymptote

• If the exponent in the denominator is larger than the exponent in the numerator then the horizontal asymptote will be y=0
• As x approaches positive or negative infinity, the denominator will be much larger than the numerator thus making the overall fraction equal zero.

Example 3: 

• A function that isn't rational may have 2 horizontal asymptotes 
• one to the left and one to the right