Unscramble the values of the following limits: Note that just substituting $0$ (or 1 or $a$) for $x$ produces the expression $\frac{0}{0}$ (which is undefined!) in each case, so this never implies anything about the value of the limit, or whether it does or does not exist.
$\lim_{x\to 0}\ \frac{3x}{x}=$ $3$ $\lim_{x\to 0}\ \frac{x^2}{x}=$ $0$
$\lim_{x\to 0^+}\ \frac{x}{x^2}=$ $\infty$ $\lim_{x\to 0^-}\ \frac{x}{x^2}=$ $-\infty$
$\lim_{x\to 0}\ \frac{x}{x^2}=$ does not exist $\lim_{x\to 0^+}\ \frac{|x|}{x}=$ $1$
$\lim_{x\to 0^-}\ \frac{|x|}{x}=$ $-1$ $\lim_{x\to 0}\ \frac{|x|}{x}=$ does not exist
$\lim_{x\to 0}\ \frac{cx}{x}=$ $c$ $\lim_{x\to 1}\ \frac{(x^2-1)}{(x-1)}=$ $2$
$\lim_{x\to 1}\ \frac{(\sqrt{x}-1)}{(x-1)}=$ $\frac{1}{2}$ $\lim_{x\to a}\ \frac{(x^2-a^2)}{(x-a)}=$ $2a$
These examples show that a $\frac{0}{0}$ limit can have any value. An expression that produces $\frac{0}{0}$ when $x=a$ is substituted, is called an indeterminate form at $a$, since the limit of such an expression has not yet been found. It usually means that more work has to be done to find the value of the limit. In the case of the examples above, this is very little work, but in the case of other limits, such as $\lim_{x\to 0}\ \frac{\sin x}{x}$ some fairly ingenius argument may be required to justify what the value is (if it exists).