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

While we can't divide by zero, we can still meaningfully ask what happens to a ratio $\displaystyle{\frac{f(x)}{g(x)}}$ 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. It shouldn't be too surprising that derivatives can help us evaluate limits that look like $0/0$.

There are other variations of this rule for evaluating limits that look like $\infty/\infty$, $0 \cdot \infty$, $\infty - \infty$, $1^\infty$ or $0^0$, or for taking limits as $x \to \pm \infty$. All these limits are called indeterminate forms. The complete statement of L'Hospital's rule and its proof is at the end of this learning module. For now we'll concentrate on learning how to apply the rule with some examples.

Indeterminate Quotients: $0/0$ or $\infty/\infty$

We start with indeterminate quotients, where either the numerator and denominator are both going to zero or are both blowing up.

Indeterminate Products: $0 \cdot \infty$

Here we consider indeterminate products where one factor is going to zero and the other is blowing up.