• We will identify the vertical asymptotes and/or point(s) of discontinuity of a rational function.
• We will study the behaviour of the rational function to the left and right side of the discontinuity.
• We will introduce the language of limits to assist us in communicating our understanding of the behaviour of the function close to its vertical asymptotes.

A rational function \( y = \dfrac{g(x)}{h(x)},\ h(x) \neq 0 \) will have a vertical asymptote at \( x = a \) if \( h(a) = 0 \) and \( g(a) \neq 0 \), when the function is in simplest form.

Consider the function \( f(x) = \dfrac{2x}{x - 5} \).

The domain of the function is \( \{x \mid x \neq 5,\ x \in \mathbb{R}\} \).

Observe that \( f(5) = \dfrac{2(5)}{5 - 5} = \dfrac{10}{0} \) which is an undefined value. The graph of the function is discontinuous at \( x = 5 \).

What happens to the value of the function (the value of \( y \)) as the value of \( x \rightarrow 5 \)?

This table shows, as \(x\) approaches \(5\) from the left, that is from numbers less than \(5\), \(y\) approaches a large negative value \((y \rightarrow -\infty)\).

Consider the function \( f(x) = \dfrac{2x}{x - 5} \).

The domain of the function is \( \{x \mid x \neq 5,\ x \in \mathbb{R}\} \).

Observe that \( f(5) = \dfrac{2(5)}{5 - 5} = \dfrac{10}{0} \) which is an undefined value. The graph of the function is discontinuous at \( x = 5 \).

This table shows, as \(x\) approaches \(5\) from the right, that is from numbers greater than \(5\), \(y\) approaches a large positive value \((y \rightarrow \infty)\).

Consider the function \( f(x) = \dfrac{2x}{x - 5} \).

The domain of the function is \( \{x \mid x \neq 5,\ x \in \mathbb{R}\} \).

Observe that \( f(5) = \dfrac{2(5)}{5 - 5} = \dfrac{10}{0} \) which is an undefined value. The graph of the function is discontinuous at \( x = 5 \).

The function \( f(x) = \dfrac{2x}{x - 5} \) has a vertical asymptote of \( x = 5 \). The graph of the function will approach, but never touch this line.