Determine the limit:
The value of the limit is (type infinity for or -infinity for )
The geometric significance of this result is that the line is a vertical asymptote for the graph of the function
Determine when a limit is infinite.
Infinite limits have a direct connection to vertical asymptotes.
In the next few examples, we will investigate infinite limits of rational functions. These typically occur at points where the denominator of the rational function is approaching zero, but the numerator is not approaching zero, leading to the undefined fraction form
example 1 Compute the limit:
Plugging in yields the expression which is undefined. As the denominator heads to zero, it gets smaller and smaller and so it will divide into the numerator more and more times. This causes the fraction to “blow up”. In terms of the limit, it is therefore reasonable to expect an answer of either or . To determine which, we do a sign analysis as follows. Since the values of are positive as , the values of are also positive.
Hence, we can conclude that
The geometric significance of this result is that the line (the -axis) is a vertical asymptote for the graph of the function , as shown below.
(problem 1)
In general, when the undefined fraction form arises in a one-sided limit, the answer
is typically and a sign analysis of the denominator will determine which of these is
correct.
example 2 Determine the limit:
First we observe that plugging in the value gives which is undefined and the fraction is “blowing up” in the limit. A sign analysis will tell us if the limit is . Since , we have and hence . Since the numerator is positive and the denominator is approaching through negative values, the values of in this limit are negative. We conclude that
This geometric significance of the result is that the line is a vertical asymptote for the graph of the function , as shown below.
(problem 2)
Determine the limit:
The value of the limit is (type infinity for or -infinity for )
The geometric significance of this result is that the line is a vertical asymptote for the graph of the function
example 3
Analyze the two-sided limit:
Plugging into the rational function gives the undefined expression . From this information, we can conclude that the one-sided limits as approaches 2 will give either or , i.e., and To determine which, we will do a sign analysis on each one-sided limit. Consider the left hand limit first: The numerator is approaching 3, which is positive. The denominator is approaching zero which is neither positive nor negative, but since , we know that and therefore . Hence the denominator is negative in this limit. Since a positive divided by a negative is negative, we get:
Now, we will do a sign analysis on the right hand limit: The numerator is approaching 3, which is positive. In the denominator, since , we know that and therefore . Hence the denominator is positive in this limit. Since a positive divided by a positive is positive, we get:
The one-sided limits were different, so the two-sided limit does not exist: The line is a vertical asymptote for the graph of , as shown below.
(problem 3)
Analyze the two-sided limit:
The value of the limit is (type infinity for , -infinity for or DNE)The geometric significance of this result is that the line is a vertical asymptote for the graph of the function
example 4 Analyze the two-sided limit:
Plugging into the rational function gives the undefined expression . From this information, we can conclude that the one-sided limits will give either or , i.e., and To determine which, we will do a sign analysis on each of the one-sided limits. Consider the left hand limit first: The numerator is approaching , which is negative. In the denominator, since , we know that and therefore . Since is negative, so is . Negative divided by negative is positive, so we get:
Now, we will do a sign analysis on the right hand limit: The numerator is still approaching . In the denominator, since , we know that and therefore . Since is positive, so is . Negative divided by positive is negative, so we get:
The one-sided limits were different, so the two-sided limit does not exist:
The graph of has a vertical asymptote at , as shown below.
The functions and also have infinite limits.
(problem 5a)
Use the result of example 5 to determine the limit:
The value of the limit is (type infinity for , -infinity for or DNE)(problem 5b)
Use the result of example 5 to determine the limit:
The value of the limit is (type infinity for , -infinity for or DNE)example 6 The graph of has a vertical asymptotes at multiples of and we have the
following limits: and
as shown in the graph below.
(problem 6a)
Compute the limit:
The value of the limit is (type infinity for , -infinity for or DNE)
(problem 6b)
Compute the limit:
The value of the limit is (type infinity for , -infinity for or DNE)(problem 6c)
Compute the limit:
The value of the limit is (type infinity for , -infinity for or DNE)
Here is a detailed, lecture style video on infinite limits:
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