https://file.aiquickdraw.com/mathbot/file/1726303119201yw1glkke.jpg

The problem shown in the image involves solving a limit:

\[
\lim_{{x \to 0}} \frac{{(x - 2)^2 - 4}}{x}
\]

Let's go step by step through the solution:

1. **Rewrite the numerator:**
   \[
   (x - 2)^2 - 4 = (x^2 - 4x + 4) - 4
   \]
   This simplifies to:
   \[
   x^2 - 4x
   \]

2. **Substitute this into the limit expression:**
   \[
   \lim_{{x \to 0}} \frac{{x^2 - 4x}}{x}
   \]

3. **Factor the numerator:**
   \[
   \frac{{x(x - 4)}}{x}
   \]

4. **Simplify by canceling the \(x\):**
   \[
   x - 4
   \]

5. **Now, substitute \(x = 0\) into the simplified expression:**
   \[
   0 - 4 = -4
   \]

Therefore, the solution is:

\[
\lim_{{x \to 0}} \frac{{(x - 2)^2 - 4}}{x} = -4
\]

Would you like more details on any of the steps? Here are 5 related questions:

1. What happens if you don't factor the numerator?
2. Why does canceling \(x\) in the limit require careful attention?
3. How would the solution change if the limit was \(x \to 2\)?
4. What if the numerator was a cubic expression?
5. How can you apply L'Hôpital's Rule to this limit?

**Tip:** Always simplify the expression as much as possible before substituting into a limit to avoid indeterminate forms like \(\frac{0}{0}\).

What is the limit of (x-2)^2 - 4 as x approaches 0, divided by x?

Learn how to solve a limit problem involving factoring: (x-2)^2 - 4 as x approaches 0, divided by x. This solution covers core concepts in algebra and limits, suitable for Grades 10-12.

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