Explains how to find asymptotes, with illustrations and examples for all 4 types of asymptotes: vertical, horizontal, skewed and asymptotic curve.

## What is an asymptote?

An asymptote is a function that another function approaches without limit as the distance from the coordinate origin increases. Put simply: An asymptote is a line that a curve approaches, as it heads towards infinity.

## What types of asymptotes are there?

• Vertical asymptote (special case, because it is not a function!)
• Horizontal asymptote
• Skewed asymptote
• Asymptotic curve

## How to find asymptotes:Vertical asymptote

A vertical asymptote (i.e. an asymptote parallel to the y-axis) is present at the point where the denominator is zero.

Therefore the calculation is easy, just calculate the zero (s) of the denominator, at that point is the vertical asymptote.

### example

The vertical asymptote of this function is to be determined:

The vertical asymptote is at the zero of the denominator, so:

So the vertical asymptote is at x = 2.

Here you can see the vertical asymptote (red) and the function (blue):

## How to find asymptotes: Skewed asymptote

This exists when the numerator degree is exactly 1 greater than the denominator degree. To calculate the asymptote, do the following:

1. Divides the numerator by the denominator and calculates this using the  polynomial division .
2. Then leave out the residual term, the result is the skewed asymptote.

### example

Compute the skewed asymptote of this function:

Perform the polynomial division, dividing the numerator by the denominator:

The blue circled is then your crooked asymptote and the orange end is the residual term, which you can then omit (always the one where the x is in the denominator). So the equation of the skewed asymptote looks like this:

The function and the skewed asymptote then look like this:

## How to find asymptotes: Horizontal asymptote

A horizontal asymptote is present in two cases:

1. When the  numerator degree is less than the denominator degree . In this case the x-axis is the horizontal asymptote
2. When the  numerator degree is equal to the denominator degree  . Then the horizontal asymptote can be calculated by dividing the factors before the highest power in the numerator by the factor of the highest power in the denominator.

### example

The horizontal asymptote of this function is sought. (Numerator degree = denominator degree)

The asymptote is then at the y-value, which results when the factors are divided before the common highest power.

This function and asymptote then look like this:

## How to find asymptotes: Asymptotic curve

This exists when the numerator degree is more than 1 greater than the denominator degree (i.e. when the numerator degree> denominator degree + 1). An asymptotic curve is an asymptote that is not a straight line, but a curve, e.g. a parabola that the graph is getting closer and closer to.

To calculate the asymptote, you proceed in the same way as for the crooked asymptote:

1. Divides the numerator by the denominator and calculates this using the  polynomial division .
2. Then leave out the remainder term (i.e. the one where the remainder stands by the denominator), the result is then the skewed asymptote.

### example

The asymptotic curve is sought for the following function (denominator degree by 2 smaller than the numerator degree, so there is an asymptotic curve):

Performs the polynomial division:

The red is then the equation of the asymptote , you can omit the part with the x in the denominator, this is the so-called residual term . So the equation of the asymptote is:

This function and asymptote looks like this:

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