End Behavior and Local Behavior of Rational Functions

Let's begin by looking at the reciprocal function, $f(x)=\frac{1}{x}$. We cannot divide by zero, which means the function is undefined at $x=0$; so zero is not in the domain. As the input values approach zero from the left side (becoming very small, negative values), the function values decrease without bound (in other words, they approach negative infinity). We can see this behavior in Table 2.

Table 2

We write in arrow notation

$\text { as } x \rightarrow 0^{-}, f(x) \rightarrow-\infty$

As the input values approach zero from the right side (becoming very small, positive values), the function values increase without bound (approaching infinity). We can see this behavior in Table 3.

Table 3

We write in arrow notation

$\text { as } x \rightarrow 0^{+}, f(x) \rightarrow \infty$

See Figure 2.

Figure 2

This behavior creates a vertical asymptote, which is a vertical line that the graph approaches but never crosses. In this case, the graph is approaching the vertical line $x=0$ as the input becomes close to zero. See Figure 3.

Figure 3

VERTICAL ASYMPTOTE

A vertical asymptote of a graph is a vertical line $x=a$ where the graph tends toward positive or negative infinity as the inputs approach $a$. We write

$\text { As } x \rightarrow a, f(x) \rightarrow \infty, \text { or as } x \rightarrow a, f(x) \rightarrow-\infty$