Suppose that \(\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0\). If

\(\lim_{x \to a} \dfrac{f'(x)}{g'(x)} = L\),

then

\(\lim_{x \to a} \dfrac{f(x)}{g(x)} = L\),

L'Hopital's rule can also be used for one-side limits and limits as \(x\) approaches \(\infty\) or \(-\infty\). And it can be also used if we find that the limit of \(f'(x)/g'(x)\) is \(\infty\) or \(-\infty\); in this case, once again, the same conclusion applies to the limit of \(f(x)/g(x)\). (You can see example 4.8.4 for a sample)

Suppose that \(\lim_{x \to a} f(x) = \infty\) or \(\lim_{x \to a} f(x) = -\infty\), and also either \(\lim_{x \to a} g(x) = \infty\) or \(\lim_{x \to a} g(x) = -\infty\). If

\(\lim_{x \to a} \dfrac{f'(x)}{g'(x)} = L\),

then

\(\lim_{x \to a} \dfrac{f(x)}{g(x)} = L\),

As before, the same rule applies to one sided limits and limits as \(x\) approaches \(\infty\) or \(-\infty\), and it also applies if \(L\) is replaced by either \(\infty\) or \(-\infty\).

Suppose that \(f\) and \(g\) are continous on \([a,b]\) and differentiable on \((a,b)\) and for all \(x \in (a,b), g'(x) \neq 0\). There there is a number \(c\) such that \(a < c < b\) and

\(\dfrac{f(b)-f(a)}{g(b)-g(a)} = \dfrac{f'(c)}{g'(c)}\)

Notice that if \(g(x) = x\) then \(g'(x) = 1\) and the above equation becomes the original mean value theorem you saw previously in section 2.