It is not difficult to find questions about evaluating limits without the use of l'Hôpital's rule. As long as the function is differentiable, not directly reducible and tends towards an intermediate form, why would anyone want to avoid such a useful tool?
Personally, I'm against any calculus technique that can be applied without (much) thinking. These days calculus (in North America, at least) is taught in a way that people can get high grades without having the slightest idea of what a derivative or an integral is. In most classes I teach I ask what an integral is, and very rarely do I get satisfactory answers, even from good students. Part of the problem is the lack of basic skills: most students are hopeless when dealing with inequalities, which prevents you from both explaining the definition of limit and doing things like Taylor polynomials.
The way I was taught calculus a million years ago, was to use Taylor (as opposed to L'Hôpital) for limits. Using Taylor approximations to find the limit allows you to have some understanding of what is going on, in particular in the sense that you are not only finding the limit but also estimating the rate of convergence. This is essential if you are doing numerical analysis, and good knowledge in any case.
This conveys more information, makes you think instead of blindly applying a formula, and avoids mistakes like the frequent one of applying L'Hôpital when it is not applicable.
In some circumstances questions like "How do I do X without Y" are genuinely intellectual exercises in working without powertools, but in other circumstances they seem more like "I have an aversion to thinking about Y so let's just do it another way."
My impression is that the first group is by far the bigger group in general, but for l'Hopital's rule specifically, it might be a mix.
Anyhow, this idea of not relying on a single route to a solution can be viewed as a positive development in the student's development :) Many students, when finishing a problem through whatever means, would just conclude "Welp, good thing I never have to think about that ever again! No chance that any portion of that problem would ever help out in a future problem because all math problems are totally disconnected and don't relate to each other or reality. It's not as if there are similar problems where the same approach won't work, requiring me to find an alternate path."
Ok they wouldn't think all of this consciously, but really that's how it seems they think sometimes...
Anyhow, the positive upshot is that a student who is used to/recognizes the value of finding alternate solutions will be more flexible in the long run.
This is a bit of a meta answer, but let me explain a bit of why a course might not cover l'Hôpital's rule. When I teach calculus I've skipped l'Hôpital for two reasons.
First, understanding when using it is and isn't circular is much more difficult than anything else covered in a Calc 1 class. Gerry gave a great example of a subtle circularity, but there are others. Since students won't be able to understand when they can and can't use it, they shouldn't use it at all.
Second, in my experience, learning l'Hôpital's rule causes students to forget everything else they ever learned about limits. In particular, many students will apply it to limits which are not indeterminate! Thus teaching l'Hôpital's rule causes more unlearning than learning, so I'd prefer to spend that time teaching another topic instead.
Now you might wonder why a class that never taught l'Hôpital's rule would have students who knew l'Hôpital's rule. When I've taught calculus usually a substantial portion of the class has taken a high school calculus class where they were taught l'Hôpital's rule but don't understand it. So I then have to give a brief explanation of why I'm not teaching it and why they shouldn't use it on the problems in the class.
Mainly because when you first study limits, you are not introduced to L'Hôpital's rule. Most limit questions come from beginners, who have not even studied derivatives. When the time comes to study L'Hôpital's, your interest in limits are generally boiled away. Hence, it is quite natural to see people asking limit questions without L'Hôpital's [perhaps because I have been through that stage].
And also, because many people see it as a challenge to find the limit without advanced techniques.
I'm pretty sure I've asked such a question, actually. The simple reason is that I was trying to solve an exercise, which was recommended to do in connection to a class which preceded the one where L'Hopital's rule was introduced. So I concluded that while the exercise was probably solvable using L'Hopital's rule (which I wasn't familiar with at that point), it was most likely intended to be solved by other means. And those 'other means' were what I was interested in, and not the limit itself.