https://r2.erweima.ai/mathbot/image/1726456257772g37y32vz.jpg

To find the solution to the given limit:

\[
\lim_{{x \to 2}} \frac{x^4 - 4x^2}{x^4 - 5x^3 + 6x^2}
\]

we need to perform algebraic manipulations to simplify the expression. Let's break it down:

1. **Factorize the numerator and the denominator**:

   - **Numerator**: \( x^4 - 4x^2 \)
     This can be factored as:
     \[
     x^2(x^2 - 4) = x^2(x - 2)(x + 2)
     \]

   - **Denominator**: \( x^4 - 5x^3 + 6x^2 \)
     Factor out \( x^2 \) from the entire expression:
     \[
     x^2(x^2 - 5x + 6)
     \]
     Then, factor the quadratic \( x^2 - 5x + 6 \) as:
     \[
     x^2 - 5x + 6 = (x - 2)(x - 3)
     \]

   So the denominator becomes:
   \[
   x^2(x - 2)(x - 3)
   \]

2. **Simplify the limit expression**:

   Now substitute the factored forms back into the original limit expression:
   \[
   \lim_{{x \to 2}} \frac{x^2(x - 2)(x + 2)}{x^2(x - 2)(x - 3)}
   \]

   Cancel out the common terms \( x^2(x - 2) \) from both the numerator and the denominator, leaving:
   \[
   \lim_{{x \to 2}} \frac{x + 2}{x - 3}
   \]

3. **Evaluate the limit**:

   Substitute \( x = 2 \) into the simplified expression:
   \[
   \frac{2 + 2}{2 - 3} = \frac{4}{-1} = -4
   \]

So, the correct answer is:

\[
\boxed{-4}
\]

This corresponds to the first option in the image.

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Would you like more details or have any questions about this solution? Here are five related questions to deepen understanding:

1. How do you determine when to factor a polynomial in limit problems?
2. What are the conditions for canceling terms in limit expressions?
3. Why is it important to check for a common factor in both the numerator and denominator?
4. How would this problem change if \( x \to 0 \) instead of \( x \to 2 \)?
5. How do you handle limits that result in an indeterminate form like \( \frac{0}{0} \)?

**Tip:** Always check for common factors in both the numerator and denominator to simplify limits. This can prevent indeterminate forms and make solving much easier!

In the process of finding the result of lim (x → 2) of the expression (x^4 - 4x^2)/(x^4 - 5x^3 + 6x^2), operations are performed to transform the expression and simplify it.

Learn how to solve the limit lim (x → 2) of the expression (x^4 - 4x^2)/(x^4 - 5x^3 + 6x^2) by factoring and simplifying. This detailed step-by-step solution includes key algebraic concepts such as polynomial factorization and limit laws. Suitable for students in Grades 10-12.

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