What are limits at infinity?

1 Answer
Oct 3, 2015

See the explanation below.

Explanation:

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..