Calculus AB and Calculus BC

CHAPTER 2 Limits and Continuity

D. LIMIT OF A QUOTIENT OF POLYNOMIALS

To find Image where P(x) and Q(x) are polynomials in x, we can divide both numerator and denominator by the highest power of x that occurs and use the fact that Image

EXAMPLE 18

Image

EXAMPLE 19

Image

EXAMPLE 20

Image

THE RATIONAL FUNCTION THEOREM

We see from Examples 18, 19, and 20 that: if the degree of P(x) is less than that of Q(x), then Image if the degree of P(x) is higher than that of Q(x), then Image (i.e., does not exist); and if the degrees of P(x) and Q(x) are the same, then Image where an and bn are the coefficients of the highest powers of x in P(x) and Q(x), respectively.

This theorem holds also when we replace “x → ∞” by “x → −∞.” Note also that:

(i) when Image then y = 0 is a horizontal asymptote of the graph of Image

(ii) when Image then the graph of Image has no horizontal asymptotes;

(iii) when Image is a horizontal asymptote of the graph of Image

EXAMPLE 21

Image