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We are given the following information:

\[
\lim_{x \to 2} f(x) = 7 \quad \text{and} \quad \lim_{x \to 2} g(x) = -7
\]

We are tasked with finding:

\[
\lim_{x \to 2} \left( -7f(x)^3 + 6f(x)^2 + 4f(x) + 6g(x)^3 + 10g(x) - 4x^2 - 8 \right)
\]

### Step-by-step solution:

#### Step 1: Substitute the limit values of \(f(x)\), \(g(x)\), and \(x\).
We can directly substitute \( \lim_{x \to 2} f(x) = 7 \), \( \lim_{x \to 2} g(x) = -7 \), and \( \lim_{x \to 2} x = 2 \) into the expression:

\[
-7f(x)^3 + 6f(x)^2 + 4f(x) + 6g(x)^3 + 10g(x) - 4x^2 - 8
\]

Substituting the given values:

\[
= -7(7)^3 + 6(7)^2 + 4(7) + 6(-7)^3 + 10(-7) - 4(2)^2 - 8
\]

#### Step 2: Simplify each term.

- \( 7^3 = 343 \), so \( -7(7)^3 = -7(343) = -2401 \).
- \( 7^2 = 49 \), so \( 6(7)^2 = 6(49) = 294 \).
- \( 4(7) = 28 \).
- \( (-7)^3 = -343 \), so \( 6(-7)^3 = 6(-343) = -2058 \).
- \( 10(-7) = -70 \).
- \( 2^2 = 4 \), so \( -4(2)^2 = -4(4) = -16 \).
- The constant \( -8 \) remains as is.

#### Step 3: Add up the terms.
\[
= -2401 + 294 + 28 - 2058 - 70 - 16 - 8
\]

Now, simplify:

\[
= -2401 + 294 + 28 - 2058 - 70 - 16 - 8
= -2401 - 2058 + 294 + 28 - 70 - 16 - 8
= -4459 + 350 - 94
= -4459 + 256 = -4203
\]

Thus, the value of the limit is \( \boxed{-4203} \).

Would you like further details on any step, or have any questions?

### Relative questions:
1. How are limits of composite functions evaluated?
2. How do you handle the limit of a polynomial function as \(x \to a\)?
3. What are the properties of limits used in this problem?
4. How does substitution work in limit problems like this?
5. How can we confirm that a function is continuous at a point?

### Tip:
When dealing with limits of complex expressions, remember to use the property that the limit of a sum is the sum of the limits, assuming each limit exists.

Suppose lim(x→2) f(x) = 7 and lim(x→2) g(x) = -7. Find lim(x→2) (-7f(x)^3 + 6f(x)^2 + 4f(x) + 6g(x)^3 + 10g(x) - 4x^2 - 8).

Solve the limit of the polynomial expression involving f(x) and g(x) at x=2, using substitution. Learn how to apply the limit properties and simplify each term step by step for functions f(x) = 7 and g(x) = -7.

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