It seems like you're asking to prove the limit:

$\lim_{x \to 0} |n|x| = -\infty.$

However, this expression seems a bit ambiguous. Let's break it down:

1. Absolute value: The expression contains $|n|$, which suggests $n$ might be a constant or a variable. If $n$ is a constant, then $|n|$ is just a positive number.
2. Product: The expression is $|n| \cdot |x|$, which is a product of the absolute value of $n$ and the absolute value of $x$.
3. Limit: We are trying to evaluate the behavior of this expression as $x \to 0$.

Analyzing the Limit

If $n$ is a constant and $x \to 0$, then:

$\lim_{x \to 0} |n|x| = |n| \cdot \lim_{x \to 0} |x|.$

Since $|x| \to 0$ as $x \to 0$, this means that:

$\lim_{x \to 0} |n|x| = |n| \cdot 0 = 0.$

Thus, the limit is 0, not $-\infty$.

Possible Clarification

If there is a negative sign or a different context you're referring to (such as $n$ being a variable that depends on $x$ in a certain way), please clarify the problem so I can provide a more accurate explanation.

Key Point:

For the limit of $|n|x|$ as $x \to 0$, the result is always 0.

Let me know if you meant something different or have additional context!

Related Questions:

1. How does the limit of a function involving absolute values behave as $x \to 0$?
2. What is the limit of $|x|$ as $x \to 0$?
3. How do we handle limits when constants are involved?
4. Can limits involving absolute values lead to negative infinity? Under what conditions?
5. What happens if $n$ is not a constant but a function of $x$?

Tip: Remember that limits involving absolute values often simplify, as the absolute value function essentially "removes" negative signs, making analysis easier.