Is this statement of Baby Rudin wrong ? Or was I taught wrong ??

Question

I am in High school in a book like baby rudin is completely out of my league. But in this question, a comment by

@YuiToCheng said "In order to apply L'hopital's rule, you only need to know $\lim \frac{f'(x)}{g'(x)}$ exists and $\lim g(x)=\infty$, see chapter 5 of baby rudin."

And, sure this is what the book says.

But being in high school I was only taught that to apply the L'hopital's rule. The condition required is $\lim g(x)=0 and \lim f(x)=0 $ and the function should be differentiable at that point. I was also taught that it was valued for $\frac{\infty}{\infty}$, but it was not mentioned taht it is valid for $\frac{anything}{\infty}$. The argument that my teacher made was that $\lim \frac{f(x)}{g(x)}= \lim \frac{\frac{1}{f(x)}}{\frac{1}{g(x)}}$ And as $\lim f(x) = 0 = \lim g(x) $ , $ \lim \frac{1}{f(x)} = \infty $, which makes it valid for $\frac{\infty}{\infty}$

Also I was taught that there is no need for the fact that $\lim \frac{f'(x)}{g'(x)}$ exists. Because if it doesn't and it is $\frac{0}{0} $ type, we can again apply the theorem. , that is just find $\lim \frac{f''(x)}{g''(x)}$

But baby rudin seems to contradict this.

So was I taught wrong ?? Or is it given wrong in the book ?? Please help.

Answer 1

1. I think your teacher made a computational point. If you are given a $\infty / \infty$ type, of course you could do the manipulation $$ \lim \frac {f(x)}{g(x)} = \lim \frac {1/g(x)}{1/f(x)} \stackrel {0/0}= \lim \frac {(1/g(x))'}{(1/f(x))'}, $$ which means you only need the limit $$ \lim \frac {-g'/g^2}{-f'/f^2} = \lim \frac {g' f^2}{f' g^2} $$ to exist and the computation is complete. Baby Rudin was dealing with $\infty/\infty$ without invoking the $0/0$ rule.

2. A counterexample $$ \lim_{x \to +\infty} \frac {\sin x}x = 0, $$ because $$ \left|\frac {\sin x}x\right| \leqslant \frac 1x \to 0 \quad [x \to +\infty], $$ but you cannot apply L'Hopital rule, since $$ \lim_{x \to +\infty} \frac {\cos x} 1 $$ does not exist.

Answer 2

You were not "taught wrong" about "anything/$\infty$". Merely what you were taught is not everything that is known. (But that is true of Rudin's version also.)

Yes, it is true that we need to know $\lim f'(x)/g'(x)$ exists. There are cases of form $0/0$ where $\lim f(x)/g(x)$ exists but $\lim f'(x)/g'(x)$ does not exist.