Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 1 / 9D (Analysis)

Rolle's theorem states that if a real function $h$ is continuous on $[a,b]$, differentiable on $(a,b)$, with $a<b$ and $h(a)=h(b)$, then some $c\in (a,b)$ satisfies $h^{\prime }(c)=0$.

For the two-function assertion, defineThis function has the required continuity and differentiability and satisfies $H(a)=H(b)=0$. Rolle's theorem therefore gives $c$ withThis proves the Cauchy mean value theorem even when either endpoint increment vanishes; no division was used.

Now impose the stated zero endpoint values and nonvanishing assumptions near $a$. For each $x\in (a,a+\delta )$, applying the proved identity on $[a,x]$ gives some $c_x\in (a,x)$ such thatAs $x\downarrow a$, every possible $c_x$ also tends to $a$. Explicitly, for any $\epsilon >0$, choose a right neighborhood of $a$ where $|f^{\prime }/g^{\prime }-L|<\epsilon$. If $x$ lies in it, so does $c_x$, and therefore $|f(x)/g(x)-L|<\epsilon$. This proves the endpoint zero-over-zero form of L'Hôpital's rule:No continuity of either derivative is required, only their ratio limit and the given nonvanishing conditions.

For the final limit, near zero $\cos x>0$, so the real logarithm is well defined. First apply the proved rule to $\tan x/x$ on the right: its derivative ratio is ${\sec }^{2}x\to 1$. Next apply it to $\log (\cos x)/x^2$; the derivative ratio isThe original quotient is even, so the left limit equals the right limit. Consequently