INFINITE LIMITS
In this section, we discuss functions whose values increase or decrease without bound as the independent variable gets closer and closer to a fixed number.
Consider:
-1.00000 3.00000
-0.80000 4.68750
-0.60000 8.33333
-0.40000 18.75000
-0.20000 75.00000
0.00000 undefined
0.20000 75.00000
0.40000 18.75000
0.60000 8.33333
0.80000 4.68750
1.00000 3.00000
increases without bound as approaches 0 from the left (through values less than 0), we write:
increases without bound as approaches 0 from the right (through values greater than 0), we write:
Thus,
NOTATION
As approaches , increases without bound is denoted by
And read as “the limit of as approaches is positive infinity”
NOTES:
1. is not a symbol for a real number
2. If the limit of as approaches is then in this case, the limit does not exist. But this tells the behaviour of the function as gets closer and closer to .
In analogous manner we can indicate the behaviour of a function whose function values decrease without bound where
is read as “the limit of as approaches is negative infinity”.
Where limit does not actually exist but tells the behaviour of the function values as approaches .
Consider
We can also consider one-sided infinite limits:
To solve problems involving Infinite Limits algebraically, the following additional limit theorems are used:
Limit Theorem 11:
If is any positive integer, then
Illustration
Limit Theorem 12:
If a is any real number and if and , where c is any constant not equal to 0
Then
1. if and if through positive values of ,
2. if and if through negative values of ,
3. if and if through positive values of ,...