Evaluate the following limits:

1. $\lim\limits_{x \to 1}F(x)$, where $F(x) = (x^2 - 4, 3x + 4)$.

2. $\lim\limits_{t \to \pi}G(t)$, where $G(t) = (\cos(2t), \sin(2t), 2t, \cos(t))$.

3. $\lim\limits_{x \uparrow 2}F(x)$, where $F(x) = (x, \sqrt{4 - x^2})$.

4. $\lim\limits_{t \to \infty}F(t)$, where $F(t) = \left(t^2e^{-2t}, \frac{t+1}{t-1}\right)$.

5. Let $F(t) = (\cos(t), \sin(t))$. Explain why $\lim\limits_{t \to \infty}\|F(t)\|$ exists, but $\lim\limits_{t \to \infty}F(t)$ does not exist.

Answers:

1. $(-3, 7)$
2. $(1, 0, 2\pi, -1)$
3. $(2, 0)$
4. $(0, 1)$