# limit (as *n* goes to infinity)

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.