I was doing some calculus homework and I came across with some problems. I have to find the following limits
1) $\displaystyle\lim_{x\to \frac{\pi}{2}} \frac{\sin(x)-1}{\cos(x)}$
2) $\displaystyle\lim_{x\to 0} \frac{x\cdot\sin(x)}{1-\cos(x)}$
3) $\displaystyle\lim_{x\to \infty} x\cdot\sin\left(\frac{\pi}{x}\right)$
4) $\displaystyle\lim_{x\to \frac{\pi}{2}} \frac{\cos(x)}{x-\frac{\pi}{2}}$
The thing is that I don't know how to solve them because all the things that I tried led me to an indetermination. I don't have to use derivatives or anything similar, just algebra "tricks". My intention isn´t having my homework done by somebody else, but I can´t come up with any idea.
Hint:
For (1) and (4), let $u=x-\pi/2$.
For (3), let $u=\pi/x$.
Use the identities $\sin\theta=2\sin(\theta/2)\cos(\theta/2)$ and $\cos\theta=1-2\sin^2(\theta/2)$ if necessary.