1. We are asked to evaluate the limit $$\lim_{n \to \infty} \left(1 - \arctan\left(\frac{1}{2}\right)\right)^n$$. 2. First, recall that $$\arctan\left(\frac{1}{2}\right)$$ is the angle whose tangent is $$\frac{1}{2}$$. This value is positive and less than $$\frac{\pi}{2}$$. 3. Numerically, $$\arctan\left(\frac{1}{2}\right) \approx 0.4636$$ (in radians), so: $$1 - \arctan\left(\frac{1}{2}\right) \approx 1 - 0.4636 = 0.5364$$. 4. Since the base $$0.5364$$ is between 0 and 1, raising it to the power $$n$$ as $$n$$ approaches infinity gives: $$\lim_{n \to \infty} (0.5364)^n = 0$$. 5. Therefore, the limit is: $$\boxed{0}$$. Note: The inequality $$\frac{\pi}{6} < \frac{1}{2} < \frac{\pi}{4}$$ serves as a comparison to understand the size of $$\frac{1}{2}$$ relative to some angle values but does not affect the limit result. 1. For the second expression mentioned, $$\lim_{n \to \infty} (4 - \arctan(\frac{4}{2}))$$, note that this is not a sequence but a fixed value since it does not depend on $$n$$. 2. Evaluate $$\arctan\left(\frac{4}{2}\right) = \arctan(2)$$, which is approximately 1.107 radians. 3. Then, $$4 - \arctan(2) \approx 4 - 1.107 = 2.893$$. 4. Since this expression does not depend on $$n$$, the limit as $$n \to \infty$$ is simply the value itself, $$\boxed{2.893}$$. Final answers are: - $$\lim_{n \to \infty} \left(1 - \arctan\left(\frac{1}{2}\right)\right)^n = 0$$ - $$\lim_{n \to \infty} (4 - \arctan(2)) = 2.893$$