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!