User:Rschwieb/Sandbox

Proof of l'Hôpital's rule

The following proof is due to (Taylor 1952), where a unified proof for the 0/0 and ±∞/±∞ indeterminate forms is given. Taylor notes that different proofs may be found in (Lettenmeyer 1936) and (Wazewski 1949).

Let f and g be functions satisfying the hypotheses in the General form section. Let ${\mathcal {I}}$ be the open interval in the hypothesis with endpoint c. Considering that $g'(x)\neq 0$ on this interval and g is continuous, ${\mathcal {I}}$ can be chosen smaller so that g is nonzero on ${\mathcal {I}}$.

For each x in the interval, define $m(x)=\inf {\frac {f'(\xi )}{g'(\xi )}}$ and $M(x)=\sup {\frac {f'(\xi )}{g'(\xi )}}$ as $\xi$ ranges over all values between x and c. (The symbols inf and sup denote the infimum and supremum.)

From the differentiability of f and g on ${\mathcal {I}}$, Cauchy's mean value theorem ensures that for any two distinct points x and y in ${\mathcal {I}}$ there exists a $\xi$ between x and y such that ${\frac {f(x)-f(y)}{g(x)-g(y)}}={\frac {f'(\xi )}{g'(\xi )}}$. Consequently $m(x)\leq {\frac {f(x)-f(y)}{g(x)-g(y)}}\leq M(x)$ for all choices of distinct x and y in the interval. The value g(x)-g(y) is always nonzero for distinct x and y in the interval, for if it was not, the mean value theorem would imply the existence of a p between x and y such that g' (p)=0.

The definition of m(x) and M(x) will result in an extended real number, and so it is possible for them to take on the values ±∞. In the following two cases, m(x) and M(x) will establish bounds on the ratio f/g.

Case 1: $\lim _{x\rightarrow c}f(x)=\lim _{x\rightarrow c}g(x)=0$

For any x in the interval ${\mathcal {I}}$, and point y between x and c,

$m(x)\leq {\frac {f(x)-f(y)}{g(x)-g(y)}}={\frac {{\frac {f(x)}{g(x)}}-{\frac {f(y)}{g(x)}}}{1-{\frac {g(y)}{g(x)}}}}\leq M(x)$

and therefore as y approaches c, ${\frac {f(y)}{g(x)}}$ and ${\frac {g(y)}{g(x)}}$ become zero, and so

$m(x)\leq {\frac {f(x)}{g(x)}}\leq M(x)$.

Case 2: $\lim _{x\rightarrow c}|g(x)|=\infty$

For any x in the interval ${\mathcal {I}}$, and point y between x and c,

$m(x)\leq {\frac {f(y)-f(x)}{g(y)-g(x)}}={\frac {{\frac {f(y)}{g(y)}}-{\frac {f(x)}{g(y)}}}{1-{\frac {g(x)}{g(y)}}}}\leq M(x)$.

As y approaches c, both ${\frac {f(x)}{g(y)}}$ and ${\frac {g(x)}{g(y)}}$ become zero, and therefore

$m(x)\leq \liminf _{x\rightarrow c}{\frac {f(y)}{g(y)}}\leq \limsup _{x\rightarrow c}{\frac {f(y)}{g(y)}}\leq M(x)$

(For readers skeptical about the x 's under the limit superior and limit inferior, remember that y is always between x and c, and so as x approaches c, so will y. The limsup and liminf are necessary: we may not yet write "lim" since the existence of that limit has not yet been established.)

In both cases,

$\lim _{x\rightarrow c}m(x)=\lim _{x\rightarrow c}M(x)=\lim _{x\rightarrow c}{\frac {f'(x)}{g'(x)}}=L$.

By the Squeeze theorem, the limit exists and $\lim _{x\rightarrow c}{\frac {f(x)}{g(x)}}=L$. This is the result that was to be proven.