To solve these limit problems, let's approach each of them using limit properties and known mathematical results. I'll go through them step by step. Here's how we'll tackle them:

1. $\lim_{x \to 2} \left( \frac{4 - x}{2} \right) \tan\left(\frac{\pi x}{4}\right)$

First, as $x$ approaches 2, substitute directly into the expression. We have: $\left( \frac{4 - 2}{2} \right) \tan\left(\frac{\pi \cdot 2}{4}\right) = 1 \cdot \tan\left(\frac{\pi}{2}\right)$ Since $\tan\left(\frac{\pi}{2}\right)$ is undefined (it approaches infinity), the limit does not exist in the traditional sense.

2. $\lim_{x \to 1} \left( \frac{\sin x}{\sin 1} \right)^{\frac{1}{x-1}}$

This limit can be approached by recognizing that the expression has the form $\left( \frac{\sin x}{\sin 1} \right)^{\frac{1}{x-1}}$, and as $x$ approaches 1, we have a $0^0$ ind