• tending to infinity
• tending to negative infinity

We also refer to this as limiting behavior.

Our shorthand notation for “the limiting behavior of” is $\lim \limits _{x \to \infty }$. This is placed to the left of the function.

$$\lim _{x \to -\infty } f(x)$$

We use limit notation to describe end-behavior, when the end-behavior is a constant or unbounded.

$\blacktriangleright$ We have seen limiting or end-behavior of exponential functions.

$$\lim _{x \to -\infty } 1.5^x = 0$$

as $x \to \infty$, the function values become unbounded and our notation for that looks like

$$\lim \limits _{x \to \infty } 1.5^x = \infty$$

$\blacktriangleright$ Same with logaarithms.

$$\lim \limits _{t \to \infty } \log _2(t+5) = \infty$$

$\blacktriangleright$ Some rational functions approach a constant in their end-behavior, which we see on their graphs as a horizontal asymptote.

graph of $y = f(x) = \frac {2x-4}{x+3}$

$$\lim \limits _{x \to -\infty } f(x) = 2$$

$$\lim \limits _{x \to \infty } f(x) = 2$$

Limits

Let $G(m)$ be defined by the following graph.

$$\lim _{x \to -\infty } f(x) = \answer {6}$$

$$\lim _{x \to \infty } f(x) = \answer {-4}$$

Limits

Let $T(y)$ be defined by

$$T(y) = \frac {2 \sin (3y)}{y} \, \text { defined on } \, (2, \infty )$$

$$\lim _{y \to \infty } T(y) = \answer {0}$$