https://file.aiquickdraw.com/mathbot/file/1727249742212958b4lo8.jpg

### Numerical Approach (T-Table)
Here is the T-Table showing the values of \( f(x) = \frac{|x+2| - 2}{|x|} \) for values of \(x\) approaching 0:

| \(x\)     | \(f(x)\) |
|-----------|----------|
| -0.100    | -1.0     |
| -0.010    | -1.0     |
| -0.001    | -1.0     |
|  0.001    |  1.0     |
|  0.010    |  1.0     |
|  0.100    |  1.0     |

- As \(x\) approaches 0 from the left (\(x \to 0^-\)), \(f(x)\) tends to -1.
- As \(x\) approaches 0 from the right (\(x \to 0^+\)), \(f(x)\) tends to 1.

### Graphical Approach
The graph of the function \( f(x) \) around \( x = 0 \) shows a jump discontinuity. From the left, the function approaches -1, and from the right, it approaches 1. The graph confirms that the limit does not exist at \( x = 0 \) because the left-hand limit and the right-hand limit are not equal.

### Conclusion
The limit does not exist since:

\[
\lim_{x \to 0^-} f(x) = -1 \quad \text{and} \quad \lim_{x \to 0^+} f(x) = 1
\]

If you need further clarification or details, feel free to ask!

Here are 5 related questions to expand your understanding:
1. What conditions must be met for a two-sided limit to exist?
2. How does absolute value influence the behavior of functions at critical points like 0?
3. How can we formally prove the non-existence of a limit using epsilon-delta definitions?
4. What are some other ways to calculate limits involving absolute value functions?
5. How do piecewise functions relate to the concept of one-sided limits?

**Tip:** When dealing with absolute value in limits, it's helpful to separately evaluate limits from the left and right to understand the function's behavior.

Evaluate the limit using all three methods/approaches: Analytical, Numerical (T-Table), and Graphical for the given expression lim(x -> 0) (|x+2| - 2) / |x|.

Learn how to evaluate the limit lim(x -> 0) (|x+2| - 2) / |x| using three approaches: analytical, numerical (T-Table), and graphical. Understand key concepts like limits, continuity, and discontinuities through absolute value functions. Suitable for Grades 11-12 and early college students.

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