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:

• 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.