1. L'Hopital Rule

Prove the following: Let $\displaystyle a<b$ be real numbers, let $\displaystyle f: [a,b] \to \bold{R}$ and $\displaystyle g: [a,b] \to \bold{R}$ be functions which are differentiable on $\displaystyle [a,b]$. Suppose that $\displaystyle f(a) = g(a) = 0$, that $\displaystyle g'$ is nonzero on $\displaystyle [a,b]$, and $\displaystyle \lim_{x \to a: x \in (a,b]} \frac{f'(x)}{g'(x)}$ exists and equals $\displaystyle L$. Then $\displaystyle g(x) \neq 0$ for all $\displaystyle x \in (a,b]$, and $\displaystyle \lim_{x \to a: x \in (a,b]} \frac{f(x)}{g(x)}$ exists and equals $\displaystyle L$.

Suppose that $\displaystyle g(x) = 0$ for some $\displaystyle x \in (a,b]$. Since $\displaystyle g(a) = 0$, then by Rolle's theorem we can find a $\displaystyle y \in (a,b]$ such that $\displaystyle g'(y) = 0$. But this contradicts the fact that $\displaystyle g'$ is nonzero on $\displaystyle [a,b]$. Hence $\displaystyle g(x) \neq 0$ for all $\displaystyle x \in (a,b]$.

Now to show that $\displaystyle \lim_{x \to a: x \in (a,b]} \frac{f'(x)}{g'(x)}$ exists and equals $\displaystyle L$, we can show that $\displaystyle \lim_{n \to \infty} \frac{f(x_{n})}{g(x_{n})} = L$ for any sequence $\displaystyle (x_{n})_{n=1}^{\infty}$ which converges to $\displaystyle x$. How would I show this? I am guessing that I have to create a new function, and somehow manipulate it so that $\displaystyle \frac{f(x)}{g(x)} = \frac{f'(x)}{g'(x)}$. And this is equivalent to showing that $\displaystyle \frac{f(x_{n})}{g(x_{n})} = \frac{f'(x_{n})}{g'(x_{n})}$.

2. The mean value theorem says that there is an x in the interval (a,b) such that $\displaystyle g'(x) [f(b)-f(a)] = f'(x)[g(b)-g(a)]$.

Now, we know that g' is nonzero (you assumed it), and if g(a) $\displaystyle \neq$ g(b), we can write $\displaystyle \frac{f'(x)}{g'(x)} = \frac{f(b)-f(a)}{g(b)-g(a)}$.

As we know f(a) = g(a) = 0,

$\displaystyle \frac{f'(x)}{g'(x)} = \frac{f(b)}{g(b)}$

If we take the limit of both sides, then x and b will approach the same point.

$\displaystyle \lim_{x\to x_0}\frac{f'(x)}{g'(x)} = \lim_{b\to b_0}\frac{f(b)}{g(b)} = \lim_{x\to x_0}\frac{f(x)}{g(x)}$

$\displaystyle \lim_{x\to x_0}\frac{f(x)}{g(x)} = \lim_{x\to x_0}\frac{f'(x)}{g'(x)}$

3. You could use Rolle's Theorem also right? In terms of sequences you can define a function $\displaystyle z_{n}: [a, x_{n}] \to \bold{R}$ by $\displaystyle z_{n}(x) = f(x)g(x_{n}) - g(x)f(x_{n})$. So this is continuous on the interval, and is zero at the endpoints. And $\displaystyle z'_{n}(x) = f'(x)g(x_{n})- g'(x)f(x_{n})$. Using Rolle's Theorem we get $\displaystyle \frac{f(x_{n})}{g(x_{n})} = \frac{f'(y_{n})}{g'(y_{n})}$. $\displaystyle x_n \to a$ as $\displaystyle n \to \infty$ and so $\displaystyle y_n \in (a, x_n) \to a$ as $\displaystyle n \to \infty$ by Squeeze Theorem.

And so $\displaystyle \frac{f'(x_{n})}{g'(x_{n})} \to L, \frac{f'(y_{n})}{g'(y_{n})} \to L \implies \frac{f(x_{n})}{g(x_{n})} \to L$.

4. Originally Posted by particlejohn
You could use Rolle's Theorem also right? In terms of sequences you can define a function $\displaystyle z_{n}: [a, x_{n}] \to \bold{R}$ by $\displaystyle z_{n}(x) = f(x)g(x_{n}) - g(x)f(x_{n})$. So this is continuous on the interval, and is zero at the endpoints. And $\displaystyle z'_{n}(x) = f'(x)g(x_{n})- g'(x)f(x_{n})$. Using Rolle's Theorem we get $\displaystyle \frac{f(x_{n})}{g(x_{n})} = \frac{f'(y_{n})}{g'(y_{n})}$. $\displaystyle x_n \to a$ as $\displaystyle n \to \infty$ and so $\displaystyle y_n \in (a, x_n) \to a$ as $\displaystyle n \to \infty$ by Squeeze Theorem.

And so $\displaystyle \frac{f'(x_{n})}{g'(x_{n})} \to L, \frac{f'(y_{n})}{g'(y_{n})} \to L \implies \frac{f(x_{n})}{g(x_{n})} \to L$.
But what you did here was proving Mean Value Theorem using Rolle's Theorem =)
(except you used sequences instead of function. see here)