Could you help me calculate the following limits:

$\displaystyle \lim_{x \to 0} x \left[ \frac{1}{x} \right]$

$\displaystyle \lim_{x\to 0} \frac{1-\cos x \cdot \sqrt{\cos2x} }{x^2}$

$\displaystyle \lim_{x\to 10} \frac{\log _{10}(x) - 1}{x-10}$

As to the last one I thought I could use $\displaystyle \lim\frac{log _{a}(1+\alpha)}{\alpha} = \log_ae$ but it wouldn't work