**Embrace** the **elegance** of **mathematics** as we delve into the exciting world of **absolute value limits**. This article explores the concept of **absolute value limits**, highlighting the **mathematical** rules, important **properties**, and the powerful **applications** they hold.

## Definition of Absolute Value Limits

In mathematics, the **limit** of a function as its argument or input tends towards a certain value is a fundamental concept in **calculus**. When we talk about the **limit** of an **absolute value function**, we are referring to this calculus concept but applied to a function involving **absolute values**.

The **absolute value function** is defined as **|x| = x** if **x ≥ 0** and **|x| = -x** if **x < 0**. Consequently, when we consider the **limit** of an **absolute value function**, it’s important to recognize that the function behaves differently on either side of the point where the argument is zero. This can lead to situations where the **right-hand limit** and **left-hand limit** are not the same.

The **limit** of an **absolute value function** at a certain point is the value that the function approaches as the input (or argument) approaches that point.

For example, consider the function **f(x) = |x – a|**. As **x** approaches **a** from the right **(x > a), f(x)** approaches **0**, and as **x** approaches **a** from the left **(x < a)**, **f(x)** also approaches **0**. Therefore, the **limit** of** f(x)** as **x** approaches a is **0**.

However, consider the function **g(x) = |x|**. As** x** approaches **0** from the right, **g(x)** approaches **0**, but as **x** approaches **0** from the left, **g(x)** also approaches 0. Therefore, the **limit** of** g(x)** as x approaches** 0** is **0**.

Figure-1.

**Properties ****of Absolute Value Limits**

The **limit** of an **absolute value function** shares many properties with** limits** in general. Here are some key properties of absolute value limits:

**Limit of the Absolute Value of a Function**

The **absolute value limits** of a function as **x** approaches **a**, can be written as:

`lim (x→a) |f(x)|`

This is equal to the **absolute value limit** of the function as** x** approaches **a**, if the limit exists.

**Squeeze (Sandwich) Theorem**

If a **function** is **bounded** by two other functions that approach the same **limit** as **x** approaches **a**, then the **original function** also approaches that limit. This theorem is often used when calculating the limits of **absolute value functions**.

**Right-hand and Left-hand Limits**

Because **absolute value functions** often have different behavior on either side of a certain point (most often where the function equals zero), it’s often necessary to calculate the **right-hand limit** (x approaching a from values greater than a) and the **left-hand limit** (x approaching a from values less than a) separately.

If these two** one-sided limits** are equal, then the **limit** of the function as** x** approaches **a** exists and is equal to this common value.

**Limit of the Absolute Value of a Difference**

If we have ** lim (x→a) |f(x) - g(x)| = 0,** it implies that

**.**

`lim (x→a) f(x) = lim (x→a) g(x)`

**Absolute Value and Inequalities**

If ** 0 < |x - a| < δ** (where

**δ**is a

**positive number**), this means that x is in the interval

**, excluding the point**

`(a - δ, a + δ)`

**x = a**. This property is especially useful when proving limits.

**Continuous at a Point**

If a function **f(x)** is continuous at a point **x = a**, then **|f(x)|** is also continuous at **x = a**.

It’s important to note that, in practice, finding the **limit** of **absolute value functions** often involves breaking the function into **piecewise-defined functions** based on where the absolute value’s argument is **positive** or **negative**.

**Exercise **

**Example 1**

Find ** lim (x→2) |x - 2|**.

Figure-2.

### Solution

Since `|x - 2|`

equals 0 when `x = 2`

, the limit as `x`

approaches 2 is also 0.

**Example 2**

Find ** lim (x→3) |x - 3| / (x - 3)**.

### Solution

This limit has different values from the right and from the left:

From the right (`x > 3`

), `|x - 3| = x - 3`

, so `lim (x→3+) |x - 3| / (x - 3) = 1`

.

From the left (`x < 3`

), `|x - 3| = -(x - 3)`

, so `lim (x→3-) |x - 3| / (x - 3) = -1`

.

Since the left-hand limit does not equal the right-hand limit, the limit `lim (x→3) |x - 3| / (x - 3)`

does not exist.

**Example 3**

Find ** lim (x→0) |x| / x**.

Figure-3.

### Solution

From the right (`x > 0`

), `|x| = x`

, so `lim (x→0+) |x| / x = 1`

.

From the left (`x < 0`

), `|x| = -x`

, so `lim (x→0-) |x| / x = -1`

.

Since the left-hand limit does not equal the right-hand limit, the limit `lim (x→0) |x| / x`

does not exist.

**Example 4**

Find ** lim (x→5) |x² - 25| / (x - 5)**.

### Solution

We can simplify `|x² - 25|`

to `|x - 5||x + 5|`

. Therefore, the limit expression becomes `|x + 5|`

.

So, `lim (x→5) |x² - 25| / (x - 5) = |5 + 5| = 10`

.

**Example 5**

Find ** lim (x→-3) |2x + 6| / (2x + 6)**.

### Solution

From the right (`x > -3`

), `|2x + 6| = 2x + 6`

, so `lim (x→-3+) |2x + 6| / (2x + 6) = 1`

.

From the left (`x < -3`

), `|2x + 6| = -(2x + 6)`

, so `lim (x→-3-) |2x + 6| / (2x + 6) = -1`

.

Since the left-hand limit does not equal the right-hand limit, the limit `lim (x→-3) |2x + 6| / (2x + 6)`

does not exist.

**Example 6**

Find ** lim (x→4) |x² - 16| / (x - 4)**.

### Solution

We can simplify `|x² - 16|`

to `|x - 4||x + 4|`

. Therefore, the limit expression becomes `|x + 4|`

.

So, `lim (x→4) |x² - 16| / (x - 4) = |4 + 4| = 8`

.

**Example 7**

Find ** lim (x→0) |sin(x)| / x**.

Figure-4.

### Solution

From the right (`x > 0`

), `|sin(x)| = sin(x)`

, so `lim (x→0+) |sin(x)| / x = 1`

.

From the left (`x < 0`

), `|sin(x)| = -sin(x)`

, so `lim (x→0-) |sin(x)| / x = -1`

.

Since the left-hand limit does not equal the right-hand limit, the limit `lim (x→0) |sin(x)| / x`

does not exist.

**Example 8**

Find ** lim (x→0) |x| / sin(x)**.

### Solution

From the right (`x > 0`

), `|x| = x`

, so `lim (x→0+) |x| / sin(x) = 1`

.

From the left (`x < 0`

), `|x| = -x`

, so `lim (x→0-) |x| / sin(x) = -1`

.

Since the left-hand limit does not equal the right-hand limit, the limit `lim (x→0) |x| / sin(x)`

does not exist.

**Applications**

The concept of** absolute value limits** not only holds importance within the realm of** mathematical theory**, but it also boasts applications across numerous **scientific fields**. Here are a few notable examples:

**Physics**

In physics, **absolute value limits** often come into play while evaluating certain physical quantities that cannot have a **negative magnitude**. For instance, determining the **time** at which a **particle** reaches a certain point, or studying **properties** of **wave functions** in **quantum mechanics**.

**Engineering**

In **control systems engineering**, the **limit** of **absolute values** is often used to analyze system **stability**. In **electrical engineering**, it’s used to find the maximum tolerable **threshold levels** for signals and currents.

**Computer Science**

In **algorithm complexity analysis** (Big O notation), **limits** of **absolute values** can help determine upper or lower bounds of algorithm performance. They’re also used in **computer graphics** to create certain visual effects and perform **image processing**.

**Economics**

Absolutely! In economics, **absolute value limits** are used to find **equilibrium points**, analyze trends, or determine the **sensitivity** of one **economic variable** to changes in another.

**Biology**

In **mathematical biology**, these limits can model **population dynamics** where the populations must remain **non-negative**.

**Environmental Science**

The concept of **limits**, including **absolute value limits**, is used in modeling **pollutant dispersion**, **population growth**, **resource management**, and more.

**Statistics and Data Analysis**

The **absolute value limit** is used when dealing with **absolute differences**, establishing **thresholds**, or defining metrics like **Mean Absolute Error** (MAE).

*All images were created with GeoGebra.*