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.