A limit "at infinity" of a function is: a number that #f(x)# (or #y#) gets close to as #x# increases without bound.

A limit at infinity is a limit as the independent variable increases without bound.

The definition is:

#lim_(xrarroo)f(x) = L# if and only if: for any #epsilon# that is positive, there is a number #m# such that: if #x > M#, then #abs(f(x)-L) < epsilon#.

For example as #x# increases without bound, #1/x# gets closer and closer to #0#.

Example 2: as #x# increases without bound, #7/x# gets closer to #0#

As #xrarroo# (as #x# increases without bound),

#(3x-2)/(5x+1) rarr 3/5#

Why?

#underbrace((3x-2)/(5x+1) = (x(3-2/x))/(x(5+1/x)))_("for " x != 0) = (3-2/x)/(5+1/x)#

As #x# increases without bound, the values of #2/x# and #1/x# go to #0#, so the expression above goes to #3/5#.

A limit "at minus infinity" of function #f#, is a number that #f(x)# approaches as #x# decreases without bound.

**Note about "without bound"**

The numbers #1/2, 3/4, 7/8, 15/16. 31/32# are increasing, but they will never get beyond #1#. The list is **bounded**

In "limits at infinity" we are interested in what happens to #f(x)# as #x# increase, but not with a bound on the increase..