Once again, we return to the example of the function:  In this function we have two examples of limits involving infinity.
First, we observe:	Here infinity is involved as we find the limit of the function as x approaches zero from the left. In reality, when the answer to a limit problem is infinity, we are really saying that there is no limit. The negative infinity answer does tell us that the value of the function is an extremely large number. This is better than simply saying there is no limit. Note: when  then the graph of the function must have a vertical asymptote.
We also see that:	In this case, we see that the right-hand limit as x approaches zero is positive infinity. We now know that officially:  because the lef-hand and right-hand limits do not agree.
We can also observe that:	Here is another way that infinity is dealt with in limits. We can take limits as . When taking a limit of the type: and the answer is a constant. Then the function has a horizontal asymptote at the function value.

A polynomial behaves like its term of highest degree as . This can be seen in the following example:
 

The  overpowers everything else in this expression as x approaches infinity. Therefore, the limit of this expression must be infinity.

When taking the limit of a rational expression and the highest power of the numerator is the same as the highest power of the denominator, then the limit of the expression as x approaches infinity is the ratio of the coefficients of the hightest degree terms in the numerator and denominator.
 

In this limit problem, the highest power term in the numerator is  and the highest power term in the denominator is . Since these are the highest power terms, they dominate the limit problem, and we can ignore the other terms in determining the limit. We see that the limit is the quotient of the coefficients of the highest power terms.

When taking the limit of a rational expression and the highest power of either the numerator or denominator is larger than the highest power of the other, then follow this advice:
 
 

	Because the numerator's largest power is larger than the denominator's largest power, the numerator's highest powered term takes this expression over as x approaches infinity. Therefore the limit of this type of expression must be positive infinity.
	Because the denominator's largest power is larger than the numerator's largest power, the denominator's highest powered term takes this expression over as x approaches inifinity. The denominator becomes larger much quicker than the denominator, therefore the limit of this expression must be zero. Because zero is a constant, this means that there is a horizontal asymptote at .

Infinite Limits



Limits at Infinity



Limits of Special Trigonometric Functions

Check Concepts

#1: True or False: A function that has a constant limit as  has a horizontal asymptote.	Choose One True False
#2: True or False: If a limit is equal to infinity, then there really is no limit.	Choose One True False
#3: True or False: If a limit is equal to infinity as x approaches 2, then at 2 the function must have a vertical asymptote.	Choose One True False
#4: In a rational function, if the highest power of the numerator and denominator are equal, then the limit of the function is equal to ______________________.	Choose One The leading coefficient of the numerator The leading coefficient of the denominator The ratio of the two coefficients There is no limit in this case
#5 True or False: If a limit is equal to zero, this means that there is no limit.	Choose One True False