1. **State the problem:** Find the limit as $x$ approaches infinity of the function $4x^2 - 7$. 2. **Recall the limit rule for polynomials:** For a polynomial $ax^n + \dots$, as $x \to \infty$, the term with the highest power dominates. 3. **Apply the rule:** Since $4x^2$ is the highest power term, as $x \to \infty$, $4x^2 \to \infty$. 4. **Evaluate the limit:** $$\lim_{x \to \infty} (4x^2 - 7) = \lim_{x \to \infty} 4x^2 - \lim_{x \to \infty} 7 = \infty - 7 = \infty$$ 5. **Interpretation:** The function grows without bound as $x$ becomes very large. **Additional note on differentiation:** Given $f$ and $g$ differentiable at $x=2$, the derivative of $f(x) - 8g(x)$ at $x=2$ is $$\frac{d}{dx}[f(x) - 8g(x)]\bigg|_{x=2} = f'(2) - 8g'(2)$$ This follows from the linearity of differentiation. **Final answer:** $$\lim_{x \to \infty} (4x^2 - 7) = \infty$$