https://file.aiquickdraw.com/mathbot/file/1728212644245oe3n72xv.jpg

The image shows the absolute value function \( y(x) = |x| \) and a related limit expression.

### 1. Absolute Value Function
The equation \( y(x) = |x| \) defines the absolute value of \( x \). The absolute value function returns the distance of \( x \) from zero, regardless of the direction on the number line, meaning:
- \( |x| = x \) if \( x \geq 0 \)
- \( |x| = -x \) if \( x < 0 \)

### 2. Limit of the Absolute Value Function
The limit expression is:
\[
\forall a \in \mathbb{R}, \lim_{x \to a} |x| = |a|
\]
This states that for any real number \( a \), as \( x \) approaches \( a \), the absolute value of \( x \) approaches the absolute value of \( a \). 

**Explanation:**
- The absolute value function is continuous everywhere, meaning that the limit of \( |x| \) as \( x \to a \) is simply \( |a| \). 
- This is true whether \( a \) is positive, negative, or zero.

For example:
- If \( a = 2 \), then \( \lim_{x \to 2} |x| = |2| = 2 \).
- If \( a = -3 \), then \( \lim_{x \to -3} |x| = |-3| = 3 \).

Would you like further details or explanations? 

Here are 5 relative questions to deepen your understanding:
1. How do we compute the derivative of the absolute value function?
2. What does it mean for a function to be continuous at a point?
3. How does the limit behave for more complex functions involving absolute values?
4. How do we interpret the limit \( \lim_{x \to 0} |x| \) geometrically?
5. Can you extend the absolute value function to higher dimensions?

**Tip:** The absolute value function is non-differentiable at \( x = 0 \), so take care when dealing with its derivatives!

Learn about the absolute value function y(x) = |x| and its limit properties. This includes the concept of limits and the continuity of the function, with examples like lim_{x -> a} |x| = |a|. Ideal for students in Grades 10-12.

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