https://file.aiquickdraw.com/mathbot/file/1726025136714how6lcp1.jpg

The limit provided in the image is:

\[
\lim_{{x \to 0^+}} \left( \frac{6}{x} - \frac{6}{|x|} \right)
\]

Let's break this down:

1. As \( x \to 0^+ \), this implies we are approaching 0 from the positive side. For \( x > 0 \), we know that \( |x| = x \) because the absolute value of a positive number is just the number itself.

2. Therefore, as \( x \to 0^+ \), the expression becomes:

\[
\lim_{{x \to 0^+}} \left( \frac{6}{x} - \frac{6}{x} \right)
\]

3. Simplifying this:

\[
\frac{6}{x} - \frac{6}{x} = 0
\]

Thus, the limit is:

\[
\boxed{0}
\]

Feel free to ask if you'd like further details or clarification!

Here are 5 related questions to deepen your understanding:
1. What happens if we change the limit direction to \( x \to 0^- \)?
2. How does the absolute value function behave for negative values of \( x \)?
3. Can you evaluate similar limits involving the absolute value function?
4. What is the general approach for solving limits involving absolute value expressions?
5. What would happen if the terms inside the limit were more complex, like involving powers or roots?

**Tip:** Always remember to analyze how the absolute value function behaves based on whether \( x \) approaches from the positive or negative side.

Find the limit if it exists: lim (x -> 0^+) (6/x - 6/|x|).

Explore how to solve the limit problem lim (x -> 0^+) (6/x - 6/|x|) with a detailed solution. Understand the behavior of absolute value and limit properties, suitable for high school students in Grades 10-12.

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