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..