## Calculus AB and Calculus BC

## CHAPTER 2 Limits and Continuity

### D. LIMIT OF A QUOTIENT OF POLYNOMIALS

To find 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

**EXAMPLE 18**

**EXAMPLE 19**

**EXAMPLE 20**

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 if the degree of *P*(*x*) is higher than that of *Q*(*x*), then (i.e., does not exist); and if the degrees of *P*(*x*) and *Q*(*x*) are the same, then where *a _{n}* and

*b*are the coefficients of the highest powers of

_{n}*x*in

*P*(

*x*) and

*Q*(

*x*), respectively.

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

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

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

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

**EXAMPLE 21**