1. **State the problem:** Find the limit as $x$ approaches infinity of the function $3x^4 - 7x^3 + 10$. 2. **Recall the rule for limits at infinity of polynomials:** The term with the highest power of $x$ dominates the behavior of the polynomial as $x \to \infty$. 3. **Identify the highest power term:** The highest power term is $3x^4$. 4. **Evaluate the limit:** Since $x^4$ grows without bound as $x \to \infty$, and the coefficient is positive, the limit is $$\lim_{x \to \infty} 3x^4 - 7x^3 + 10 = \lim_{x \to \infty} 3x^4 = \infty.$$ 5. **Conclusion:** The limit diverges to infinity because the highest degree term dominates and grows without bound.