12.4 Limits at Infinity

Definition of Limits at Infinity:
If f is a function and the limits are real numbers, then
  and
 
These limits are read as "the limit of f(x) as x approaches is "
and "the limit of f(x) as x approaches is ."


You can use the properties of limits to evaluate a limit at infinity.
Example:





So, the limit of as x approaches is 4.


When f(x) is a rational function, and the largest term in the numerator and denominator have the same exponential value for the variable, the limit can be reduced to the coefficients of those terms.
Example:

This complicated limit can be reduced to

because you can ignore everything but the largest terms in the numerator and denominator and reduce the fraction that's left.

Evaluating Limits

Limits of Polynomial and Rational Fuctions

1.       If p is a polynomial function and c is a real number, then

     

     2. If r is a rational function given by                                                                and c is a real number, then





To find the limit of a polynomial or rational function, you can do a few things.

The first is direct substitution. This means that you take the number that x is approaching and substitute it in for x in F(x).

Example:

7(-3) +12

-9

The second is by dividing out. This means that you have to factor the numerator and denominator and divide out the common factors, and then use direct substitution.

Example:

(x+1)

1+1

2

The third is the rationalizing technique. This is when you first rationalize the numerator of a function.

Example:

Using direct substitution, this would be indeterminate, or
  However, you can rewrite the fraction by rationalizing the numerator.

 Now the problem can be solved by direct substitution. when 0 is put into the equation for x, the limit can be determined to be negative one half.

-Rachel