Find limits at infinity.

End Behavior

In this section we consider limits as approaches either or . There is a connection to the value of these limits and horizontal asymptotes.

Compute the limit:
As goes to infinity, so does
As the denominator grows, the fraction shrinks
The fraction never shrinks below 0
The value of the limit is

An important generalization of the last example and problem, which follows from a similar analysis is for any constant, , and any positive exponent, .

We will exploit this important fact in the next two examples.

Compute the limit:
When you ‘plug in’ , you get an form
Determine the highest power of in the denominator
Factor this power from both the numerator and denominator and cancel it
as .
The value of the limit is

Compute the limit:
When you ‘plug in’ , you get an form
Determine the highest power of in the denominator
Factor this power from both the numerator and denominator and cancel it
as .
The value of the limit is

Compute the limit:
When you ‘plug in’ , you get an form
Determine the highest power of in the denominator
Factor this power from both the numerator and denominator
Cancel the power of that was factored out
as .
The value of the limit is

Compute the limit:
When you ‘plug in’ , you get an form
Determine the highest power of in the denominator
Factor this power from both the numerator and denominator
Cancel the power of that was factored out
as .
The value of the limit is

Compute the limit:
When you ‘plug in’ , you get an form
Determine the highest power of in the denominator
Factor this power from both the numerator and denominator
Cancel the power of that was factored out
as .
The value of the limit is (type infinity for , -infinity for or DNE)

Compute
as

The value of the limit is (type infinity for , -infinity for or DNE)

Compute
oscillates between and with period

The value of the limit is (type infinity for , -infinity for or DNE)

Here is a detailed, lecture style video on finding limits at infinity:
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