Vertical and Horizontal Asymptotes Revision Notes for HSC SSCE Mathematics Advanced

Vertical and Horizontal Asymptotes

When sketching unknown functions, we follow several important steps. After factoring, we:

Identify the domain
Test whether the function has even or odd symmetry, or neither
Identify the zeroes and discontinuities and create a table of signs
Identify the curve's vertical and horizontal asymptotes

This note focuses on the fourth step: identifying asymptotes and describing how the curve behaves near them.

What are asymptotes?

Asymptotes commonly appear in functions involving algebraic fractions, such as:

$y = \frac{2}{3-x}$ or $y = \frac{1}{4-x^2}$ or $y = \frac{x-1}{x-4}$

In these functions, the denominator becomes very small when $x$ is near certain values. For example, the denominator is very small when $x$ is near $3$ for the first function, or near $2$ or $-2$ for the second function, or near $4$ for the third function. In all three functions, the denominator becomes very large when $x$ is very large.

infoNote

Asymptotes occur naturally when working with algebraic fractions. They appear in two situations:

When the denominator approaches zero (creating vertical asymptotes)
When x becomes very large in magnitude (potentially creating horizontal asymptotes)

Understanding asymptotes relies on two key principles about reciprocals:

chatImportant

Principle 1: Size relationships

The reciprocal of a very small number is a very large number
The reciprocal of a very large number is a very small number

Principle 2: Sign relationships

The reciprocal of a positive number is positive
The reciprocal of a negative number is negative

The rectangular hyperbola

These principles are clearly demonstrated by the rectangular hyperbola $y = \frac{1}{x}$ , which has both a vertical and a horizontal asymptote.

Horizontal behaviour: When $x$ is a very large number (positive or negative), $y$ is a very small number with the same sign. This means the curve approaches the $x$ -axis on both the left and right sides. We write this as:

"As $x \to \infty$ , $y \to 0$ , and as $x \to -\infty$ , $y \to 0$ "

Vertical behaviour: When $x$ is a very small number (positive or negative), $y$ is a very large number with the same sign. This means the curve flies off to $\infty$ or to $-\infty$ near $x = 0$ . We write this as:

"As $x \to 0^-$ , $y \to -\infty$ , and as $x \to 0^+$ , $y \to +\infty$ "

infoNote

The notation $x \to 0^-$ means "as $x$ approaches $0$ from the left", and $x \to 0^+$ means "as $x$ approaches $0$ from the right". This precise notation helps us describe the curve's behavior on each side of the asymptote.

In more complex situations, we use a table of signs to check the sign and distinguish between $y \to \infty$ and $y \to -\infty$ .

Testing for vertical asymptotes

When finding vertical asymptotes, always factor the function first as far as possible.

chatImportant

Key rule for vertical asymptotes:

If the denominator has a zero at $x = a$ , and the numerator is not zero at $x = a$ , then the vertical line $x = a$ is an asymptote.

To determine whether $y \to \infty$ or $y \to -\infty$ , construct a table of signs. Once you've identified the vertical asymptote, you can describe the curve's behaviour near it using the notation $x \to a^+$ and $x \to a^-$ .

Example: Finding vertical asymptotes

Let's examine the function $y = \frac{2}{3-x}$ .

lightbulbExample

Worked Example: Finding Vertical Asymptote for $y = \frac{2}{3-x}$

Finding the vertical asymptote:

When $x = 3$ , the denominator equals zero but the numerator does not. Therefore, x = 3 is a vertical asymptote.

Constructing a table of signs:

There are no zeroes of the function, but there is a discontinuity at $x = 3$ .

$x$	$0$	$3$	$4$
$y$	$\frac{2}{3}$	$*$	$-2$
sign	$+$	$*$	$-$

From the table of signs:

As $x \to 3^+$ , $y \to -\infty$
As $x \to 3^-$ , $y \to +\infty$

The horizontal asymptote:

The $x$ -axis is a horizontal asymptote because as $x \to \infty$ , $y \to 0$ , and as $x \to -\infty$ , $y \to 0$ .

The sketch:

Example: Function with two vertical asymptotes

Consider the function $f(x) = \frac{1}{4-x^2}$ .

lightbulbExample

Worked Example: Function with Two Vertical Asymptotes $f(x) = \frac{1}{4-x^2}$

Testing for symmetry:

$f(-x) = \frac{1}{4-(-x)^2} = \frac{1}{4-x^2} = f(x)$

Therefore, the function is even (it has symmetry about the $y$ -axis).

Finding vertical asymptotes:

Factoring: $y = \frac{1}{(2-x)(2+x)}$

When $x = 2$ or $x = -2$ , the denominator equals zero but the numerator does not. Therefore, x = 2 and x = -2 are both vertical asymptotes.

Constructing a table of signs:

There are no zeroes, and there are discontinuities at $x = 2$ and $x = -2$ .

$x$	$-3$	$-2$	$0$	$2$	$3$
$y$	$-\frac{1}{5}$	$*$	$\frac{1}{4}$	$*$	$-\frac{1}{5}$
sign	$-$	$*$	$+$	$*$	$-$

From the table of signs:

As $x \to (-2)^-$ , $y \to -\infty$ , and as $x \to (-2)^+$ , $y \to +\infty$
As $x \to 2^-$ , $y \to +\infty$ , and as $x \to 2^+$ , $y \to -\infty$

The horizontal asymptote:

The $x$ -axis is a horizontal asymptote: as $x \to \infty$ , $y \to 0$ , and as $x \to -\infty$ , $y \to 0$ .

Horizontal asymptotes and behaviour for large $x$

For simple functions like those above, identifying the $x$ -axis as a horizontal asymptote is straightforward. However, for more complex functions such as:

$y = \frac{x-1}{x-4}$ or $y = \frac{x-1}{x^2-4}$

finding horizontal asymptotes requires a systematic approach. These functions are called rational functions because they are the ratio of two polynomials.

Method for finding horizontal asymptotes

chatImportant

Method for Finding Horizontal Asymptotes:

Step 1: Divide both the numerator and denominator by the highest power of x in the denominator.

Step 2: Use the fact that $\frac{1}{x} \to 0$ as $x \to \infty$ and as $x \to -\infty$ .

Step 3: If $f(x)$ tends to a definite limit $b$ as $x \to \infty$ or as $x \to -\infty$ , then the horizontal line $y = b$ is a horizontal asymptote on the right or on the left.

Example: Rational function with horizontal asymptote $y = 1$

Examine the function $y = \frac{x-1}{x-4}$ .

lightbulbExample

Worked Example: Finding Horizontal Asymptote for $y = \frac{x-1}{x-4}$

Finding the horizontal asymptote:

Divide both numerator and denominator by $x$ :

$y = \frac{1 - \frac{1}{x}}{1 - \frac{4}{x}}$

As $x \to \infty$ : $y \to \frac{1-0}{1-0} = 1$

As $x \to -\infty$ : $y \to \frac{1-0}{1-0} = 1$

Therefore, y = 1 is a horizontal asymptote.

Finding the vertical asymptote:

When $x = 4$ , the denominator equals zero but the numerator does not. Therefore, x = 4 is a vertical asymptote.

Table of signs:

We need to check around the zero at $x = 1$ and the discontinuity at $x = 4$ .

$x$	$0$	$1$	$2$	$4$	$5$
$y$	$\frac{1}{4}$	$0$	$-\frac{1}{2}$	$*$	$4$
sign	$+$	$0$	$-$	$*$	$+$

From the table:

As $x \to 4^-$ , $y \to -\infty$
As $x \to 4^+$ , $y \to +\infty$

The sketch:

Example: Different types of behaviour

Examine the behaviour of these functions as $x \to \infty$ and as $x \to -\infty$ .

lightbulbExample

Worked Example: Analyzing Different Asymptotic Behaviors

a) $y = \frac{x-1}{x^2-4}$

Divide numerator and denominator by $x^2$ :

$y = \frac{\frac{1}{x} - \frac{1}{x^2}}{1 - \frac{4}{x^2}}$

As $x \to \infty$ : $y \to \frac{0-0}{1-0} = 0$

As $x \to -\infty$ : $y \to \frac{0-0}{1-0} = 0$

The $x$ -axis y = 0 is a horizontal asymptote.

b) $y = \frac{x^2-1}{x-4}$

Divide numerator and denominator by $x$ :

$y = \frac{x - \frac{1}{x}}{1 - \frac{4}{x}}$

As $x \to \infty$ : $y \to \frac{\infty - 0}{1-0} = \infty$

As $x \to -\infty$ : $y \to \frac{-\infty - 0}{1-0} = -\infty$

There are no horizontal asymptotes.

c) $y = \frac{3-5x-4x^2}{4-5x-3x^2}$

Divide numerator and denominator by $x^2$ :

$y = \frac{\frac{3}{x^2} - \frac{5}{x} - 4}{\frac{4}{x^2} - \frac{5}{x} - 3}$

As $x \to \infty$ : $y \to \frac{0-0-4}{0-0-3} = \frac{4}{3}$

As $x \to -\infty$ : $y \to \frac{0-0-4}{0-0-3} = \frac{4}{3}$

Therefore, y = 4/3 is a horizontal asymptote.

d) $y = 2^x + \frac{1}{x^2+1} + 3$

As $x \to \infty$ : $y \to \infty + 0 + 3 = \infty$

As $x \to -\infty$ : $y \to 0 + 0 + 3 = 3$

Therefore, y = 3 is a horizontal asymptote on the left only.

infoNote

Notice how the degree of the numerator compared to the denominator determines asymptotic behavior:

When the numerator has lower degree, the $x$ -axis is a horizontal asymptote (example a)
When the numerator has higher degree, there are no horizontal asymptotes (example b)
When numerator and denominator have equal degree, there's a horizontal asymptote at the ratio of leading coefficients (example c)

Remember!

bookmarkSummary

Key Points to Remember:

Vertical asymptotes occur where the denominator equals zero but the numerator does not. Always factor the function first.
Use tables of signs to determine whether the function approaches $+\infty$ or $-\infty$ near vertical asymptotes.
For horizontal asymptotes, divide the numerator and denominator by the highest power of x in the denominator, then examine what happens as $x \to \pm\infty$ .
Reciprocal principles: Small denominators create large fractions, and large denominators create small fractions. The sign of the reciprocal matches the sign of the original number.
Limit notation like $x \to a^+$ (approaching from the right) and $x \to a^-$ (approaching from the left) precisely describes behaviour near asymptotes.

Vertical and Horizontal Asymptotes (HSC SSCE Mathematics Advanced): Revision Notes