1. **State the problem:** Evaluate the expression $$\int_0^3 1 \, dx + \lim_{n \to \infty} \left(3 - \frac{1}{n}\right) - 3 \over \cos(0)$$. 2. **Evaluate the integral:** The integral of 1 with respect to $x$ from 0 to 3 is $$\int_0^3 1 \, dx = [x]_0^3 = 3 - 0 = 3.$$ 3. **Evaluate the limit:** $$\lim_{n \to \infty} \left(3 - \frac{1}{n}\right) = 3 - \lim_{n \to \infty} \frac{1}{n} = 3 - 0 = 3.$$ 4. **Substitute values into the numerator:** $$3 + 3 - 3 = 3.$$ 5. **Evaluate the denominator:** $$\cos(0) = 1.$$ 6. **Form the fraction and simplify:** $$\frac{3}{1} = 3.$$ **Final answer:** $$3$$.