I'm numbering you're problems:
1. 2.
3. 4.
5. 6.
7.
1.) Note that $\displaystyle \frac{x}{|x|}=sign(x)=\left\{\begin{array}{lr}1&:x >0\\0&:x=0\\-1&:x<0\end{array}\right\}$
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2.) This function is continuous (from the right) at $\displaystyle x=1$.
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3.) Just plug it in.
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4.) Start by factoring the numerator and see where that gets you.
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5.) Same strategy as the last one.
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6.) As $\displaystyle x$ gets small, $\displaystyle \frac{1}{|x|}$ gets really big, so $\displaystyle \lim_{x\to0}\frac{1}{|x|}=\infty$.
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7.) Same method as (4) and (5).