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The given expression is:

\[
\lim_{{x \to 2}} \frac{x^3 - 2x^2}{x^2 - 4}
\]

Let's solve this step by step.

### Step 1: Substitute \( x = 2 \) directly into the expression.
If we substitute \( x = 2 \) directly into the expression:

\[
\frac{(2^3 - 2 \times 2^2)}{(2^2 - 4)} = \frac{(8 - 8)}{(4 - 4)} = \frac{0}{0}
\]

We get the indeterminate form \( \frac{0}{0} \). This means we need to apply algebraic simplification or use L'Hôpital's Rule.

### Step 2: Factor the denominator and numerator.
Factor the denominator first:

\[
x^2 - 4 = (x - 2)(x + 2)
\]

Next, factor the numerator:

\[
x^3 - 2x^2 = x^2(x - 2)
\]

### Step 3: Simplify the expression.
Now rewrite the limit with the factored terms:

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

We can cancel out the common factor \( (x - 2) \) from both the numerator and the denominator:

\[
\lim_{{x \to 2}} \frac{x^2}{x + 2}
\]

### Step 4: Substitute \( x = 2 \) into the simplified expression.
Now that we have eliminated the indeterminate form, we substitute \( x = 2 \) into the simplified expression:

\[
\frac{2^2}{2 + 2} = \frac{4}{4} = 1
\]

### Final Answer:
\[
\lim_{{x \to 2}} \frac{x^3 - 2x^2}{x^2 - 4} = 1
\]

Would you like further details or explanations?

Here are 5 related questions to expand on this concept:
1. What is L'Hôpital's Rule, and when is it used to solve limits?
2. How do you factor polynomials like \( x^3 - 2x^2 \)?
3. What other techniques can be used to simplify indeterminate forms like \( \frac{0}{0} \)?
4. Can limits involving higher-order polynomials also be simplified by factoring?
5. How does the behavior of limits help in understanding continuity of functions?

**Tip:** Always check for the possibility of factoring polynomials when you encounter indeterminate forms like \( \frac{0}{0} \) in limits.

Learn how to solve the limit problem lim(x -> 2) (x^3 - 2x^2) / (x^2 - 4) by simplifying an indeterminate form using factoring techniques. This detailed solution is suitable for high school students learning about limits and indeterminate forms.

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