lim x --> negative 2^+(x-2)/(|x|-2) solve

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.

Explore how to solve the limit lim_{x \to -2^+} \frac{x - 2}{|x| - 2} step by step, including handling absolute values and division by zero. This detailed solution explains the behavior of limits as x approaches -2 from the right. Suitable for students in Grade 11-12 or college-level calculus.

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