
Glossary: Prime Pages: Top 5000: 
Roughly, "L is the limit of f(n) as n
goes to infinity" means "when n gets big,
f(n) gets close to L." So, for example,
the limit of 1/n is 0. The limit of sin(n)
is undefined because sin(n) continues to oscillate
as x goes to infinity, it never approaches any
single value.
Technically, L is the limit of f(n) as n goes to infinity if and only if for every e>0, there is a b>0 such that f(n)L<e whenever n>b. Let's apply this to our first example above (the limit of 1/n is 0): Suppose e is any positive number. If we let b=1/e, then n>b is 1/n<e. This is enough to prove the limit of 1/n is 0. Look in the front of almost any Calculus textbook to see more examples of this definition, as well as the numerous other forms of limits.
Chris K. Caldwell © 19992017 (all rights reserved)
