The problem from the image reads as follows:

$A = \lim_{x \to 3} \frac{3(x^2 - 9)(x^2 + 7)}{(x^2 - 2x)}.$

Step-by-Step Solution:

1. Simplify the numerator:

   • The numerator is $3(x^2 - 9)(x^2 + 7)$.
   • Notice that $x^2 - 9$ can be factored as: $x^2 - 9 = (x - 3)(x + 3).$
   • So, the numerator becomes: $3(x - 3)(x + 3)(x^2 + 7).$

2. Simplify the denominator:

   • The denominator is $x^2 - 2x$.
   • Factor out $x$ from $x^2 - 2x$: $x^2 - 2x = x(x - 2).$

3. Substitute the factored forms:

   • Rewrite the expression: $A = \lim_{x \to 3} \frac{3(x - 3)(x + 3)(x^2 + 7)}{x(x - 2)}.$

4. Handle the indeterminate form:

   • If we substitute $x = 3$ directly, the term $x - 3$ in the numerator becomes $0$, leading to a division by zero. This means the limit results in an indeterminate form, so we need to simplify further.

5. Cancel out the common factor $x - 3$:

   • The term $x - 3$ is common in the numerator and denominator. Cancel it out (noting that $x \neq 3$ while taking the limit): $A = \lim_{x \to 3} \frac{3(x + 3)(x^2 + 7)}{x(x - 2)}.$

6. Substitute $x = 3$:

   • After cancellation, substitute $x = 3$ into the simplified expression: $A = \frac{3((3) + 3)((3)^2 + 7)}{(3)((3) - 2)}.$
   • Simplify each term:
     • $3 + 3 = 6$,
     • $3^2 + 7 = 9 + 7 = 16$,
     • $3 - 2 = 1$.
   • Substitute these values: $A = \frac{3(6)(16)}{3(1)}.$

7. Simplify the final expression:

   • The factor $3$ cancels out from the numerator and denominator: $A = 6 \cdot 16 = 96.$

Final Answer:

$A = 96.$

Would you like further clarification or additional details?

5 Related Questions:

1. What are some techniques to evaluate limits involving indeterminate forms?
2. How can factoring simplify limits involving polynomials?
3. What is L'Hôpital's Rule, and when is it applicable?
4. Can you explain why we cancel terms like $x - 3$ in limits even though $x = 3$ is undefined?
5. How do we interpret the meaning of limits in calculus?

Tip:

Always check for common factors to simplify a limit before applying other techniques, as it often avoids unnecessary complexity!