How to apply limit properties in this case? I was asked to find the following limit:
$$
\lim_{x \to 1} \frac{1 - \sqrt x}{1 - x}
$$
I worked it out using direct substitution that the limit is $\frac{1}{2}$.
Initially I was trying a more algebraic approach, finding separately the limits of 
$$
\lim_{x \to 1} 1 - \sqrt x = 0
$$
$$
\lim_{x \to 1} 1 - x = 0
$$
And then applying limit properties for division and multiplication: 
$$
\lim_{x \to a} (f \cdot g)(x) = l \cdot m
$$
$$
\lim_{x \to a} (\frac{1}{g})(x) = \frac{1}{m}
$$
But that doesn't work, since the limit in the denominator will $0$. 
So besides direct substitution, the limit properties are of no use in this case when there is a $0$ in the denominator?

Since the function involves a square root of $x$ we tacitly assume $x >0$ and from $x\to 1$ we assume that $x\neq 1$. Then we may use $1- x = (1 - \sqrt x)(1 + \sqrt x)$. It follows that
$$
\lim_{x \to 1} \frac{1 - \sqrt x}{1 - x} = \lim_{x \to 1} \frac{1 - \sqrt x}{(1 - \sqrt x)(1 + \sqrt x)} = \lim_{x \to 1} \frac{1}{1 + \sqrt x} = \frac{1}{2}
$$
