The Limit does not exist or does it
- Slides: 13
The Limit does not exist. . . or does it? Section 13 -4 Limits of Infinite Sequences
Objective: To find or estimate the limit of an infinite sequence or to determine that “the limit does not exist. ”
A series that does not have a last term is called infinite. Examine the following sequence: How small can the terms get? As x is replaced with increasingly larger values, what happens? Obviously, the terms are getting smaller. Is there a number that the sequence approaches? Let’s look at the graph. 0, 6 The graph is 0, 5 asymptotically 0, 4 approaching the x 0, 3 0, 2 -axis which means 0, 1 the sequence 0 0 2 4 6 8 10 approaches zero!
What are the possibilities? • The sequence approaches a number (as in the previous example. • The sequence can approach infinity or negative infinity. • The sequence has no limit. Clarification: In order for a sequence to have a limit, the sequence must approach a single value or either positive or negative infinity.
The Limit Does Not Exist? Separate the sequence: The positive odd numbers approach infinity. The negative even numbers approach negative infinity. Since they do not approach the same thing, there is no limit. The limit does not exist!
Theorem: What this means is that any fraction having x to a positive power in the denominator (power in the denominator > power of x in the numerator) will approach zero!! This provides an important method for calculating limits: if presented with a fraction, divide the numerator and denominator by the highest power of x in the denominator.
Example 1: The highest power in the denominator is n 2, so divide each term in the numerator and the denominator by n 2. 0 0 When n is very large 1/n 2 and 3/n are near zero.
Example 2: The highest power in the denominator is n, so divide each term in the numerator and the denominator by n. 0 When n is very large 1/n is near zero while n is near .
Example 3: The highest power in the denominator is n, so divide each term in the numerator and the denominator by n. 0 When n is very large 1/n is near zero while n is near .
Example 4: The highest power in the denominator is n 2, so divide each term in the numerator and the denominator by n 2. 0 When n is very large 1/n is near zero while n is near .
Example 5: The highest power in the denominator is n, so divide each term in the numerator and the denominator by n. 0 When n is very large 1/n is near zero.
Example 6: The highest power in the denominator is n 2, so divide each term in the numerator and the denominator by n 2. 0 When n is very large 1/n 2 is near zero.
Example 7: The highest power in the denominator is n, so divide each term in the numerator and the denominator by n. 0 0 When n is very large 1/n is near zero.