## Polynomial functions and rational functions.

Proposition 4.5.1   In a neighborhood of (resp. ), a polynomial function has the same limit as its term with greatest power.

Proof. Let , where all th 's are real and . Then we have:

When gets either positive or negative with arbitrary large absolute value, every fraction has limit equal to 0, thus the sum in parentheses has limit equal to 1, whence the result.

Example 4.5.2   For , and .

The same kind of computation can be useful for non polynomial expressions, as shown in the next example.

Example 4.5.3   Let . We compute the limit of at :

Thus .

Proposition 4.5.4   In a neighborhood of (resp. ), a rational function has the same limit as the quotient of the term with greatest power in the numerator by the term with greatest power in the denominator.

The proof is a prototype of the computations that we will make later, for solving other cases where both the numerator and the numerator have an infinite limit at some point .

Proof. Let be a rational function, say

where and . We push out the greatest power of in the numerator and the greatest power of in the numerator:

Now every fraction and every fraction has limit equal to 0 when gets either positive or negative with arbitrary large absolute value, therefore both expressions in parentheses have a limit at equal to 1, whence the result.

Example 4.5.5   If , then:

Noah Dana-Picard 2007-12-28