TITLE: Avoiding circular logic using L'Hospital's rule
QUESTION [5 upvotes]: Often, using L'Hospital's rule can make a limit much simpler to evaluate, but in some circumstances it can be incorrect to use the rule even when all of its criteria are met - one example being the evaluation of $\lim_{x\to0}\frac{\sin x}{x}$, which relies on $\frac{d}{dx}\sin x=\cos x$ being known, which itself relies on the limit we are trying to prove!
How, when considering using L'Hospital's rule, can examples like this one be spotted in order to avoid circular logic?
At first glance, I wouldn't have known that the proof was circular, and I'm concerned that I might make similar mistakes with other functions.

REPLY [4 votes]: This is a slightly different take on this question than the other nicely written answer by Henning Makholm. 

If you are trying to evaluate any limit of the form $$\lim_{x\to a} \frac{f(x) - f(a)} {x-a} $$ via L'Hospital's Rule then you are doing things in a pretty roundabout way. This is more like getting hold of your right ear with your left hand and that too by going through the back of your head. It is preferable to use your right hand to get hold of your right ear. Usage of L'Hospital's Rule in this case is correct only when $f'$ is continuous at $a$ and it will fail when this is not the case (try with $f(x) =x^{2}\sin(1/x),f(0)=0,a=0$).
Some people prefer to say that this kind of usage of L'Hospital's Rule is circular. This is not exactly the case because the limit in question is $f'(a) $ and L'Hospital's Rule never uses the value $f'(a) $, but rather it uses values of $f'(x) $ for $x\neq a$ and then we take limit of $f'(x) $ as $x\to a$. In most common cases the value $f'(x) $ is obtained by a formula which is also applicable when $x=a$ and hence it just makes more sense to use that formula to calculate $f'(a) $ rather than evaluating the limit of $f'(x) $ as $x\to a$.
For some specific functions $f$ it may be possible that the formula for $f'(x) $ for general $x$ is dependent on the evaluation of $f'(a) $ and in these cases the use of L'Hospital's Rule is definitely circular. Thus if the function $f(x) = \sin x$ is defined using geometrical definition then the evaluation of the derivative of $\sin x$ is crucially dependent on the limit formula $\lim_{x\to 0}\dfrac{\sin x} {x} =1$ and then the usage of L'Hospital's Rule for evaluation of this limit is indeed circular.