The given limit problem is:

$\lim_{{x \to -2^+}} \frac{{x - 2}}{{|x| - 2}}$

Let's break it down step by step:

Step 1: Understanding the problem

You are asked to find the limit as $x \to -2^+$, which means $x$ approaches $-2$ from the right (slightly greater than $-2$).

Step 2: Analyze the expression

The expression is $\frac{x - 2}{|x| - 2}$, where:

• $x - 2$ is the straightforward difference.
• $|x| - 2$ involves the absolute value of $x$.

Step 3: Evaluate the absolute value at $x = -2^+$

When $x$ approaches $-2$ from the right, we have:

• $|x| = -x$ because $x$ is negative for values near $-2$, and the absolute value of a negative number is positive.

Thus, $|x| = -x$ when $x < 0$.

Step 4: Simplifying the expression near $x = -2^+$

For $x \to -2^+$, substitute $|x| = -x$, so the expression becomes:

$\frac{{x - 2}}{{|x| - 2}} = \frac{{x - 2}}{{-x - 2}}$

Now, simplify this fraction:

$\frac{{x - 2}}{{-x - 2}} = \frac{(x - 2)}{-(x + 2)}$

Step 5: Find the limit

Now, evaluate the limit as $x \to -2^+$:

$\lim_{{x \to -2^+}} \frac{{x - 2}}{{-(x + 2)}}$

Substituting $x = -2$ directly into the expression:

$\frac{{-2 - 2}}{{-(-2 + 2)}} = \frac{{-4}}{{0}}$

This results in a division by zero, which suggests that the limit tends towards infinity or negative infinity.

Step 6: Determine the behavior of the expression

• As $x \to -2^+$, the numerator $(x - 2)$ approaches $-4$.
• The denominator $-(x + 2)$ approaches $0^+$ (since $x + 2 \to 0^+$).

Thus, the overall expression behaves as $\frac{-4}{0^+} = -\infty$.

Conclusion:

$\lim_{{x \to -2^+}} \frac{{x - 2}}{{|x| - 2}} = -\infty$

Would you like further details or clarifications?

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Here are 5 related questions to explore further:

1. How does the behavior of limits change when approaching from the left vs. the right?
2. What happens when you take the limit $x \to 2^+$ for the same function?
3. How does the presence of absolute value impact the evaluation of limits?
4. Can you analyze the limit if $x \to -2^-$ instead?
5. What are some techniques to handle limits with division by zero?

Tip: Always check the direction of approach in limits (left or right) as it significantly affects the result.