What does $\frac{0}{0}$ mean?

While we can't divide by zero, we can still meaningfully ask what happens to a ratio $\frac{f(x)}{g(x)}$ when both $f(x)$ and $g(x)$ go to zero.  In fact, this sort of ratio appears every time we take a derivative: $$ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}.$$ We can't just plug in $h=0$, since then both the numerator and denominator would be zero.  Derivatives can help us evaluate limits that look like $\frac{0}{0}$.

In addition, limits that look like $0 \cdot \infty$, $\infty - \infty$, $1^\infty$, $\infty^0$, or $0^0$ can be manipulated to use L'Hospital's rule.  All these limits are called indeterminate forms, which means that the limit value may be $0,\infty,-\infty,54,-100$ or anything.  We can tell nothing without doing more work.  We have seen some indeterminate forms in the past, and now we have another tool to use when we do more work.

Let us look at these ideas, and see intuitively why L'Hospital's rule works.