The limit expression you provided is:

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

Let's break this down step by step.

Step 1: Understand the expression

• The term $|n|$ represents the absolute value of $n$, which is a constant. So, $|n|$ is a positive real number.
• The term $|x|$ is the absolute value of $x$, which means $|x| \geq 0$ for all $x$.

So the expression simplifies to:

$|n| \cdot |x|.$

This is the product of a positive constant $|n|$ and the absolute value of $x$.

Step 2: Examine the behavior as $x \to 0$

As $x \to 0$, $|x|$ approaches 0. Therefore, the expression $|n| \cdot |x|$ also approaches 0. Since $|n|$ is constant and positive, the limit becomes:

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

Step 3: Conclusion

The limit is actually 0, not $-\infty$. Therefore, the statement provided in the question is incorrect.

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

Recap:

• The correct limit is $0$, not $-\infty$.
• The given expression might have a typo or misunderstanding, as there is no reason why the absolute value of $x$ would lead to a limit of $-\infty$.

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Would you like further clarification or details? Here are some related questions for deeper exploration:

1. What happens if the expression involves a negative constant instead of an absolute value, like $-n \cdot x$?
2. How do limits behave for products of constants and functions near 0?
3. How does the absolute value affect the behavior of a limit as $x$ approaches 0?
4. Can limits involving absolute values lead to nonzero finite results when approaching 0?
5. How do you handle indeterminate forms like $0 \cdot \infty$ in limits?

Tip: When evaluating limits involving absolute values, remember that the absolute value function changes the direction of negative values but does not affect the magnitude, which can help simplify the evaluation.