The function you're referring to is:

$\frac{x}{|x|}$

This is the ratio of $x$ to its absolute value. Let's analyze the behavior of this function as $x$ approaches 0 from both the left and the right.

As $x \to 0^-$ (approaching 0 from the left):

• When $x$ is negative, $|x| = -x$, so: $\frac{x}{|x|} = \frac{x}{-x} = -1$ Therefore, as $x$ approaches 0 from the left, the expression approaches $-1$.

As $x \to 0^+$ (approaching 0 from the right):

• When $x$ is positive, $|x| = x$, so: $\frac{x}{|x|} = \frac{x}{x} = 1$ Therefore, as $x$ approaches 0 from the right, the expression approaches $1$.

Conclusion:

• As $x \to 0^-$, the expression $\frac{x}{|x|}$ approaches $-1$.
• As $x \to 0^+$, the expression $\frac{x}{|x|}$ approaches $1$.

Since the left-hand limit and the right-hand limit are not the same, the two-sided limit does not exist. This means there is no single number $L$ that the function values approach as $x \to 0$.

Additionally, the function is not defined at $x = 0$ because dividing by $|x|$ at $x = 0$ would require division by zero, which is undefined.

Your answer to fill in:

• As $x$ approaches 0 from the left, $\frac{x}{|x|}$ approaches $-1$.
• As $x$ approaches 0 from the right, $\frac{x}{|x|}$ approaches $1$.

This means there is no single number $L$ that the function values get arbitrarily close to as $x$ approaches 0.

B. The function is not defined at $x = 0$, so the limit cannot exist.

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Would you like further clarification on any of the points? Here are some related questions:

1. What does it mean for a limit to exist?
2. Can you explain why the limit of $\frac{x}{|x|}$ doesn't exist at $x = 0$?
3. How do you handle limits when the function has different values approaching from the left and the right?
4. What are other examples of functions where the limit does not exist?
5. How would the function change if the denominator were $x$ instead of $|x|$?

Tip: When dealing with absolute value in limits, always check the behavior separately for values approaching from the left and right, as it can affect the outcome.