# What are limits at infinity?

Oct 3, 2015

See the explanation below.

#### Explanation:

A limit "at infinity" of a function is: a number that $f \left(x\right)$ (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:

${\lim}_{x \rightarrow \infty} f \left(x\right) = L$ if and only if: for any $\epsilon$ that is positive, there is a number $m$ such that: if $x > M$, then $\left\mid f \left(x\right) - L \right\mid < \epsilon$.

For example as $x$ increases without bound, $\frac{1}{x}$ gets closer and closer to $0$.

Example 2: as $x$ increases without bound, $\frac{7}{x}$ gets closer to $0$

As $x \rightarrow \infty$ (as $x$ increases without bound),

$\frac{3 x - 2}{5 x + 1} \rightarrow \frac{3}{5}$

Why?

${\underbrace{\frac{3 x - 2}{5 x + 1} = \frac{x \left(3 - \frac{2}{x}\right)}{x \left(5 + \frac{1}{x}\right)}}}_{\text{for } x \ne 0} = \frac{3 - \frac{2}{x}}{5 + \frac{1}{x}}$

As $x$ increases without bound, the values of $\frac{2}{x}$ and $\frac{1}{x}$ go to $0$, so the expression above goes to $\frac{3}{5}$.

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

The numbers $\frac{1}{2} , \frac{3}{4} , \frac{7}{8} , \frac{15}{16.} \frac{31}{32}$ are increasing, but they will never get beyond $1$. The list is bounded
In "limits at infinity" we are interested in what happens to $f \left(x\right)$ as $x$ increase, but not with a bound on the increase..