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: