lim*𝑥*→−3*𝑥*2+9(*𝑥*3+3)(*𝑥*4+1)

To solve the limit \( \lim_{x \to -3} \left[ x^2 + 9(x^3 + 3)(x^4 + 1) \right] \), you can directly substitute \( x = -3 \) into the expression because it's a polynomial, and polynomials are continuous everywhere.

Let's break it down:

\[
x^2 + 9(x^3 + 3)(x^4 + 1)
\]

Substituting \( x = -3 \):

\[
(-3)^2 + 9((-3)^3 + 3)((-3)^4 + 1)
\]

Now calculate each term:

1. \( (-3)^2 = 9 \)
2. \( (-3)^3 = -27 \), so \( (-3)^3 + 3 = -27 + 3 = -24 \)
3. \( (-3)^4 = 81 \), so \( (-3)^4 + 1 = 81 + 1 = 82 \)
4. Now, multiply the terms in the expression:

\[
9(-24)(82)
\]

Calculate \( 9 \times (-24) \):

\[
9 \times (-24) = -216
\]

Now multiply by 82:

\[
-216 \times 82 = -17712
\]

Finally, add this to the first term:

\[
9 + (-17712) = -17703
\]

Thus, the limit is:

\[
\lim_{x \to -3} \left[ x^2 + 9(x^3 + 3)(x^4 + 1) \right] = -17703
\]

Would you like any further details or explanations?

Here are 5 related questions you might explore:
1. How does the degree of a polynomial affect the behavior of limits at infinity?
2. What techniques are useful for evaluating limits involving products of polynomials?
3. Can you find the derivative of \( f(x) = x^2 + 9(x^3 + 3)(x^4 + 1) \) at \( x = -3 \)?
4. How would the limit change if the expression were \( \lim_{x \to -3} \frac{x^2 + 9(x^3 + 3)(x^4 + 1)}{x + 3} \)?
5. What is the role of continuity in determining the limit of a polynomial function?

**Tip:** When evaluating limits of polynomial functions, substitution is the most straightforward method due to the continuous nature of polynomials.

Learn how to evaluate the limit of the polynomial x^2 + 9(x^3 + 3)(x^4 + 1) as x approaches -3. Detailed step-by-step solution provided. Suitable for advanced high school students and college-level calculus enthusiasts.

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