# Nakafa Learning Content

> For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`.

URL: https://nakafa.com/en/subjects/mathematics/function-modeling/asymptote
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/function-modeling/asymptote/en.mdx

Find vertical, horizontal, and oblique asymptotes with worked examples and interactive visualizations. Learn techniques for graphing rational functions.

---

## What is an Asymptote?

Have you ever noticed a function graph that approaches a line but never touches it? Well, that line is called an **asymptote**!

An asymptote is a straight line that is approached by a function graph when its variable value approaches infinity or approaches a certain value. Imagine like you're walking towards a wall but never actually touching it, that's the concept of an asymptote.

## Types of Asymptotes

There are three types of asymptotes you need to know:

### Vertical Asymptote

A vertical asymptote is a vertical line that the graph approaches when the function value approaches positive or negative infinity.

**Definition:** The line $$x = a$$ is a vertical asymptote if:

Visible text: **Definition:** The line is a vertical asymptote if:

- When $$x$$ approaches $$a$$ from the left, $$f(x) \to \pm\infty$$
- When $$x$$ approaches $$a$$ from the right, $$f(x) \to \pm\infty$$

Visible text: - When approaches from the left, 
- When approaches from the right,

**How to find:** For rational functions, vertical asymptotes occur when $$\text{denominator} = 0$$ and $$\text{numerator} \neq 0$$, or when $$Q(x) = 0$$ and $$P(x) \neq 0$$.

Visible text: **How to find:** For rational functions, vertical asymptotes occur when and , or when and .

### Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph approaches when $$x$$ approaches positive or negative infinity.

Visible text: A horizontal asymptote is a horizontal line that the graph approaches when approaches positive or negative infinity.

**Definition:** The line $$y = b$$ is a horizontal asymptote if:

Visible text: **Definition:** The line is a horizontal asymptote if:

- $$\lim_{x \to \infty} f(x) = b$$
- $$\lim_{x \to -\infty} f(x) = b$$

Visible text: - 
-

### Oblique Asymptote (Oblique)

An oblique asymptote is a slanted line that the graph approaches when $$x$$ approaches infinity.

Visible text: An oblique asymptote is a slanted line that the graph approaches when approaches infinity.

**Definition:** The line $$y = mx + c$$ is an oblique asymptote if:

Visible text: **Definition:** The line is an oblique asymptote if:

```math
\lim_{x \to \pm\infty} [f(x) - (mx + c)] = 0
```

## Asymptotes in Rational Functions

Let's focus on rational functions $$f(x) = \frac{P(x)}{Q(x)}$$ where $$P(x)$$ and $$Q(x)$$ are polynomials.

Visible text: Let's focus on rational functions where and are polynomials.

### Finding Vertical Asymptotes

**Steps:**

1. Find the value of $$x$$ that makes $$Q(x) = 0$$
2. Check if $$P(x) \neq 0$$ at that value
3. If yes, then there is a vertical asymptote at $$x = a$$

Visible text: 1. Find the value of that makes 
2. Check if at that value
3. If yes, then there is a vertical asymptote at

**Example:** Determine the vertical asymptote of $$f(x) = \frac{x + 3}{x - 2}$$

Visible text: **Example:** Determine the vertical asymptote of

**Solution:**

- Denominator is zero when: $$x - 2 = 0$$, so $$x = 2$$
- When $$x = 2$$, numerator is $$2 + 3 = 5 \neq 0$$
- Therefore, vertical asymptote: $$x = 2$$

Visible text: - Denominator is zero when: , so 
- When , numerator is 
- Therefore, vertical asymptote:

Let's look at the function behavior around the vertical asymptote:

| $$x$$    | $$f(x) = \frac{x + 3}{x - 2}$$                       | Description                              |
| -------------------------- | ---------------------------------------------------------------------- | ---------------------------------------- |
| $$1.9$$  | $$\frac{1.9 + 3}{1.9 - 2} = \frac{4.9}{-0.1} = -49$$ | Approaches $$-\infty$$ |
| $$1.99$$ | $$\frac{4.99}{-0.01} = -499$$                        | Getting more negative                    |
| $$2.01$$ | $$\frac{5.01}{0.01} = 501$$                          | Approaches $$+\infty$$ |
| $$2.1$$  | $$\frac{5.1}{0.1} = 51$$                             | Getting more positive                    |

Visible text: | | | Description |
| -------------------------- | ---------------------------------------------------------------------- | ---------------------------------------- |
| | | Approaches |
| | | Getting more negative |
| | | Approaches |
| | | Getting more positive |

Component: LineEquation
Props:
- title: Graph of $$f(x) = \frac{x + 3}{x - 2}$$ with Vertical
Asymptote
  Visible text: Graph of with Vertical
Asymptote
- description: Notice how the graph approaches the vertical line{" "}
$$x = 2$$ without ever touching it.
  Visible text: Notice how the graph approaches the vertical line{" "}
 without ever touching it.
- data: [
{
points: [
{ x: -2, y: 0.25, z: 0 },
{ x: -1, y: 0.67, z: 0 },
{ x: 0, y: 1.5, z: 0 },
{ x: 1, y: 4, z: 0 },
{ x: 1.5, y: 9, z: 0 },
{ x: 1.8, y: 29, z: 0 },
{ x: 1.9, y: 49, z: 0 },
],
color: getColor("PURPLE"),
showPoints: false,
labels: [{ text: "x → 2^{-}", at: 1, offset: [-1, 0.5, 0] }],
},
{
points: [
{ x: 2.1, y: 51, z: 0 },
{ x: 2.2, y: 31, z: 0 },
{ x: 2.5, y: 11, z: 0 },
{ x: 3, y: 6, z: 0 },
{ x: 4, y: 3.5, z: 0 },
{ x: 5, y: 2.67, z: 0 },
{ x: 6, y: 2.25, z: 0 },
],
color: getColor("PURPLE"),
showPoints: false,
labels: [{ text: "x → 2^{+}", at: 6, offset: [0.5, -0.5, 0] }],
},
{
points: [
{ x: 2, y: -50, z: 0 },
{ x: 2, y: 0, z: 0 },
{ x: 2, y: 50, z: 0 },
],
color: getColor("ORANGE"),
showPoints: false,
labels: [{ text: "x = 2", at: 1, offset: [1, -0.5, 0] }],
},
]
- cameraPosition: [10, 6, 10]
- showZAxis: false

### Finding Horizontal Asymptotes

**Rules for rational functions:**

Let the degree of numerator is $$m$$ and degree of denominator = $$n$$

Visible text: Let the degree of numerator is and degree of denominator =

1. If $$m < n$$: Horizontal asymptote is $$y = 0$$
2. If $$m = n$$: Horizontal asymptote is $$y = \frac{a}{b}$$ (ratio of leading coefficients)
3. If $$m > n$$: No horizontal asymptote (but there might be an oblique asymptote)

Visible text: 1. If : Horizontal asymptote is 
2. If : Horizontal asymptote is (ratio of leading coefficients)
3. If : No horizontal asymptote (but there might be an oblique asymptote)

**Example:** Determine the horizontal asymptote of:

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

   **Solution:**

   The numerator degree is $$= 1$$ and the denominator degree is $$= 2$$. Since $$1 < 2$$, the horizontal asymptote is $$y = 0$$.

2. $$g(x) = \frac{3x^2 - 1}{2x^2 + 5}$$

   **Solution:**

   The numerator degree is $$= 2$$ and the denominator degree is $$= 2$$. Since the degrees are equal, the horizontal asymptote is $$y = \frac{3}{2}$$.

Visible text: 1. 

 **Solution:**

 The numerator degree is and the denominator degree is . Since , the horizontal asymptote is .

2. 

 **Solution:**

 The numerator degree is and the denominator degree is . Since the degrees are equal, the horizontal asymptote is .

Let's see how the function approaches the horizontal asymptote:

| $$x$$    | $$g(x) = \frac{3x^2 - 1}{2x^2 + 5}$$                              | Approaches                |
| -------------------------- | ----------------------------------------------------------------------------------- | ------------------------- |
| $$10$$   | $$\frac{3(100) - 1}{2(100) + 5} = \frac{299}{205} \approx 1.459$$ | $$1.5$$ |
| $$100$$  | $$\frac{29999}{20005} \approx 1.4997$$                            | $$1.5$$ |
| $$1000$$ | $$\frac{2999999}{2000005} \approx 1.49997$$                       | $$1.5$$ |

Visible text: | | | Approaches |
| -------------------------- | ----------------------------------------------------------------------------------- | ------------------------- |
| | | |
| | | |
| | | |

Component: LineEquation
Props:
- title: Graph of $$g(x) = \frac{3x^2 - 1}{2x^2 + 5}$$ with
Horizontal Asymptote
  Visible text: Graph of with
Horizontal Asymptote
- description: The graph approaches $$y = 1.5$$ when{" "}
$$x \to \pm\infty$$.
  Visible text: The graph approaches when{" "}
.
- data: [
{
points: Array.from({ length: 40 }, (_, i) => {
const x = -10 + i * 0.5;
const y = (3 * x * x - 1) / (2 * x * x + 5);
return { x, y, z: 0 };
}),
color: getColor("TEAL"),
showPoints: false,
},
{
points: [
{ x: -10, y: 1.5, z: 0 },
{ x: 0, y: 1.5, z: 0 },
{ x: 10, y: 1.5, z: 0 },
],
color: getColor("AMBER"),
showPoints: false,
labels: [{ text: "y = 1.5", at: 1, offset: [2, 0.5, 0] }],
},
]
- cameraPosition: [10, 6, 10]
- showZAxis: false

### Finding Oblique Asymptotes

Oblique asymptotes appear when $$\text{degree of numerator} = \text{degree of denominator} + 1$$.

Visible text: Oblique asymptotes appear when .

**How to find:** Perform polynomial division.

**Example:** Determine the oblique asymptote of $$f(x) = \frac{x^2 + 2x - 1}{x - 1}$$

Visible text: **Example:** Determine the oblique asymptote of

**Solution:**
Using polynomial division:

Component: MathContainer
Children:

```math
f(x) = \frac{x^2 + 2x - 1}{x - 1} = x + 3 + \frac{2}{x - 1}
```

When $$x \to \pm\infty$$, the term $$\frac{2}{x - 1} \to 0$$

Visible text: When , the term

Therefore, oblique asymptote: $$y = x + 3$$

Visible text: Therefore, oblique asymptote:

Component: LineEquation
Props:
- title: Graph of $$f(x) = \frac{x^2 + 2x - 1}{x - 1}$$ with
Oblique Asymptote
  Visible text: Graph of with
Oblique Asymptote
- description: The graph approaches the line $$y = x + 3$$ when{" "}
$$x \to \pm\infty$$.
  Visible text: The graph approaches the line when{" "}
.
- data: [
{
points: Array.from({ length: 30 }, (_, i) => {
const x = -8 + i * 0.3;
if (Math.abs(x - 1) < 0.1) return null;
const y = (x * x + 2 * x - 1) / (x - 1);
return { x, y, z: 0 };
}).filter((p) => p !== null),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: Array.from({ length: 30 }, (_, i) => {
const x = 1.3 + i * 0.3;
const y = (x * x + 2 * x - 1) / (x - 1);
return { x, y, z: 0 };
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: -8, y: -5, z: 0 },
{ x: 0, y: 3, z: 0 },
{ x: 10, y: 13, z: 0 },
],
color: getColor("PINK"),
showPoints: false,
labels: [{ text: "y = x + 3", at: 1, offset: [3, 1.5, 0] }],
},
{
points: [
{ x: 1, y: -10, z: 0 },
{ x: 1, y: 0, z: 0 },
{ x: 1, y: 15, z: 0 },
],
color: getColor("ORANGE"),
showPoints: false,
labels: [{ text: "x = 1", at: 1, offset: [1, -0.5, 0] }],
},
]
- cameraPosition: [10, 6, 10]
- showZAxis: false

## Drawing Graphs with Asymptotes

Asymptotes are very helpful in drawing function graphs. Here are the steps:

1. **Determine all asymptotes** (vertical, horizontal, or oblique)
2. **Draw asymptotes with dashed lines**
3. **Find intercepts** with the axes
4. **Determine some additional points**
5. **Draw the curve** that approaches the asymptotes

**Complete Example:** Draw the graph of $$f(x) = \frac{x + 1}{x - 2}$$

Visible text: **Complete Example:** Draw the graph of

**Step** $$1$$: Find asymptotes

Visible text: **Step** : Find asymptotes

- Vertical asymptote: $$x = 2$$ ($$\text{denominator} = 0$$)
- Horizontal asymptote: $$y = 1$$ (same degree, coefficient ratio $$= 1/1$$)

Visible text: - Vertical asymptote: ()
- Horizontal asymptote: (same degree, coefficient ratio )

**Step** $$2$$: Intercepts

Visible text: **Step** : Intercepts

- $$y$$-axis: $$f(0) = \frac{0 + 1}{0 - 2} = -\frac{1}{2}$$
- $$x$$-axis: $$0 = \frac{x + 1}{x - 2}$$, so $$x = -1$$

Visible text: - -axis: 
- -axis: , so

**Step** $$3$$: Behavior around asymptotes

Visible text: **Step** : Behavior around asymptotes

- When $$x \to 2^-$$: $$f(x) \to -\infty$$
- When $$x \to 2^+$$: $$f(x) \to +\infty$$
- When $$x \to \pm\infty$$: $$f(x) \to 1$$

Visible text: - When : 
- When : 
- When :

**Step** $$4$$: Value table to help with drawing

Visible text: **Step** : Value table to help with drawing

| $$x$$  | $$f(x) = \frac{x + 1}{x - 2}$$                  | Description                      |
| ------------------------ | ----------------------------------------------------------------- | -------------------------------- |
| $$-3$$ | $$\frac{-3 + 1}{-3 - 2} = \frac{-2}{-5} = 0.4$$ | Point in quadrant I              |
| $$-1$$ | $$\frac{-1 + 1}{-1 - 2} = \frac{0}{-3} = 0$$    | $$x$$-axis intercept |
| $$0$$  | $$\frac{0 + 1}{0 - 2} = \frac{1}{-2} = -0.5$$   | $$y$$-axis intercept |
| $$1$$  | $$\frac{1 + 1}{1 - 2} = \frac{2}{-1} = -2$$     | Approaching vertical asymptote   |
| $$3$$  | $$\frac{3 + 1}{3 - 2} = \frac{4}{1} = 4$$       | Right of asymptote               |
| $$5$$  | $$\frac{5 + 1}{5 - 2} = \frac{6}{3} = 2$$       | Approaching horizontal asymptote |

Visible text: | | | Description |
| ------------------------ | ----------------------------------------------------------------- | -------------------------------- |
| | | Point in quadrant I |
| | | -axis intercept |
| | | -axis intercept |
| | | Approaching vertical asymptote |
| | | Right of asymptote |
| | | Approaching horizontal asymptote |

Component: LineEquation
Props:
- title: Complete Graph of $$f(x) = \frac{x + 1}{x - 2}$$
  Visible text: Complete Graph of
- description: Graph with vertical asymptote $$x = 2$$ and horizontal
asymptote $$y = 1$$.
  Visible text: Graph with vertical asymptote and horizontal
asymptote .
- data: [
{
points: Array.from({ length: 40 }, (_, i) => {
const x = -5 + i * 0.175;
if (Math.abs(x - 2) < 0.05) return null;
const y = (x + 1) / (x - 2);
if (Math.abs(y) > 20) return null;
return { x, y, z: 0 };
}).filter((p) => p !== null),
color: getColor("SKY"),
showPoints: false,
},
{
points: Array.from({ length: 40 }, (_, i) => {
const x = 2.05 + i * 0.175;
const y = (x + 1) / (x - 2);
if (Math.abs(y) > 20) return null;
return { x, y, z: 0 };
}).filter((p) => p !== null),
color: getColor("SKY"),
showPoints: false,
},
{
points: [
{ x: 2, y: -20, z: 0 },
{ x: 2, y: 0, z: 0 },
{ x: 2, y: 20, z: 0 },
],
color: getColor("ORANGE"),
showPoints: false,
labels: [{ text: "x = 2", at: 1, offset: [2, -0.5, 0] }],
},
{
points: [
{ x: -5, y: 1, z: 0 },
{ x: 0, y: 1, z: 0 },
{ x: 9, y: 1, z: 0 },
],
color: getColor("AMBER"),
showPoints: false,
labels: [{ text: "y = 1", at: 1, offset: [1, 0.5, 0] }],
},
{
points: [{ x: -1, y: 0, z: 0 }],
color: getColor("TEAL"),
showPoints: true,
labels: [{ text: "(-1, 0)", at: 0, offset: [-1, -0.5, 0] }],
},
{
points: [{ x: 0, y: -0.5, z: 0 }],
color: getColor("TEAL"),
showPoints: true,
labels: [{ text: "(0, -0.5)", at: 0, offset: [-1.5, -1, 0] }],
},
]
- cameraPosition: [10, 6, 10]
- showZAxis: false

## Practice Problems

1. Determine all asymptotes of $$f(x) = \frac{2x^2 - 3x + 1}{x - 3}$$

2. Determine the asymptotes of $$g(x) = \frac{x^2 - 4}{x^2 - 9}$$

3. The average cost function of a product is $$C(x) = \frac{500 + 3x}{x}$$. Determine the minimum cost per unit that can be achieved.

4. Draw a sketch of the graph $$h(x) = \frac{x}{x^2 - 1}$$ complete with its asymptotes.

Visible text: 1. Determine all asymptotes of 

2. Determine the asymptotes of 

3. The average cost function of a product is . Determine the minimum cost per unit that can be achieved.

4. Draw a sketch of the graph complete with its asymptotes.

### Answer Key

**Answer** $$1$$:

Visible text: **Answer** :

- $$\text{degree of numerator }(2) = \text{degree of denominator }(1) + 1$$
- There is an oblique asymptote. By division: $$f(x) = 2x + 3 + \frac{10}{x - 3}$$
- Vertical asymptote: $$x = 3$$
- Oblique asymptote: $$y = 2x + 3$$

Visible text: - 
- There is an oblique asymptote. By division: 
- Vertical asymptote: 
- Oblique asymptote:

**Answer** $$2$$:

Visible text: **Answer** :

- Vertical asymptote: $$x^2 - 9 = 0$$, so $$x = 3$$ and $$x = -3$$
- But when $$x = 2$$, $$\text{numerator} = 0$$, so $$x = 2$$ is not an asymptote
- When $$x = -2$$, $$\text{numerator} = 0$$, so $$x = -2$$ is not an asymptote
- Horizontal asymptote: $$y = 1$$ (same degree, ratio $$= 1/1$$)

Visible text: - Vertical asymptote: , so and 
- But when , , so is not an asymptote
- When , , so is not an asymptote
- Horizontal asymptote: (same degree, ratio )

**Answer** $$3$$:

Visible text: **Answer** :

```math
C(x) = \frac{500 + 3x}{x} = \frac{500}{x} + 3
```

When $$x \to \infty$$, $$\frac{500}{x} \to 0$$
So minimum cost per unit $$= 3$$

Visible text: When , 
So minimum cost per unit

**Answer** $$4$$:

Visible text: **Answer** :

- Vertical asymptotes: $$x = 1$$ and $$x = -1$$
- Horizontal asymptote: $$y = 0$$ (degree of numerator is less than degree of denominator)
- The graph has three separate parts due to two vertical asymptotes

Visible text: - Vertical asymptotes: and 
- Horizontal asymptote: (degree of numerator is less than degree of denominator)
- The graph has three separate parts due to two vertical asymptotes

Value table for $$h(x) = \frac{x}{x^2 - 1}$$:

Visible text: Value table for :

| $$x$$    | $$h(x)$$                              | Description |
| -------------------------- | ------------------------------------------------------- | ----------- |
| $$-2$$   | $$\frac{-2}{4-1} = -\frac{2}{3}$$     | Left part   |
| $$-0.5$$ | $$\frac{-0.5}{0.25-1} = \frac{2}{3}$$ | Middle part |
| $$0$$    | $$\frac{0}{0-1} = 0$$                 | Intercept   |
| $$0.5$$  | $$\frac{0.5}{0.25-1} = -\frac{2}{3}$$ | Middle part |
| $$2$$    | $$\frac{2}{4-1} = \frac{2}{3}$$       | Right part  |

Visible text: | | | Description |
| -------------------------- | ------------------------------------------------------- | ----------- |
| | | Left part |
| | | Middle part |
| | | Intercept |
| | | Middle part |
| | | Right part |

Component: LineEquation
Props:
- title: Graph of $$h(x) = \frac{x}{x^2 - 1}$$ with Two Vertical
Asymptotes
  Visible text: Graph of with Two Vertical
Asymptotes
- description: Graph with vertical asymptotes at $$x = -1$$ and{" "}
$$x = 1$$, and horizontal asymptote{" "}
$$y = 0$$.
  Visible text: Graph with vertical asymptotes at and{" "}
, and horizontal asymptote{" "}
.
- data: [
{
points: Array.from({ length: 20 }, (_, i) => {
const x = -3 + i * 0.095;
const y = x / (x * x - 1);
if (Math.abs(y) > 10) return null;
return { x, y, z: 0 };
}).filter((p) => p !== null),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: Array.from({ length: 20 }, (_, i) => {
const x = -0.95 + i * 0.09;
const y = x / (x * x - 1);
if (Math.abs(y) > 10) return null;
return { x, y, z: 0 };
}).filter((p) => p !== null),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: Array.from({ length: 20 }, (_, i) => {
const x = 1.05 + i * 0.095;
const y = x / (x * x - 1);
if (Math.abs(y) > 10) return null;
return { x, y, z: 0 };
}).filter((p) => p !== null),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: -1, y: -10, z: 0 },
{ x: -1, y: 0, z: 0 },
{ x: -1, y: 10, z: 0 },
],
color: getColor("ORANGE"),
showPoints: false,
labels: [{ text: "x = -1", at: 1, offset: [-2, 1, 0] }],
},
{
points: [
{ x: 1, y: -10, z: 0 },
{ x: 1, y: 0, z: 0 },
{ x: 1, y: 10, z: 0 },
],
color: getColor("ORANGE"),
showPoints: false,
labels: [{ text: "x = 1", at: 1, offset: [2, -0.5, 0] }],
},
{
points: [
{ x: -3, y: 0, z: 0 },
{ x: 0, y: 0, z: 0 },
{ x: 3, y: 0, z: 0 },
],
color: getC ... [truncated; 1295 chars]
- cameraPosition: [10, 6, 10]
- showZAxis: false