Q:

Calculate limite: $\lim_{x \to 2} \frac{\cos(\pi/x)}{2-x}$ without L'Hospital's rule.

I need to calculate the following limit without using L'Hospital's rule:
$$\lim_{x \to 2} \frac{\cos(\pi/x)}{2-x}$$

A:

Let $t=2-x$. Then,
$$\lim_{x \to 2} \frac{\cos(\pi/x)}{2-x}=\lim_{t \to 0} \frac{\cos\frac{\pi}{2-t}}{t}
=\lim_{t \to 0} \frac{\sin\frac{t\pi}{2(t-2)}}{t}
=\lim_{t \to 0} \frac{\sin\frac{t\pi}{2(t-2)}}{\frac{t\pi}{2(t-2)}}\lim_{t \to 0}\frac{\pi}{2(t-2)}=-\frac\pi4$$
where $\lim_{u \to 0}\frac{\sin u}{u} =1$ is used.

