The case for L'Hôpital's rule?

Its great that you raised this point as a question (+1)! The issue has regularly been discussed in comments to many questions/answers tagged limits but now there is hope that all the arguments of both types (for and against) can be found in one place.

There is a very common misconception that limits can be evaluated via plugging and use of L'Hospital's Rule even furthers this line of reasoning that limits can be evaluated by "differentiating and plugging" instead of just "plugging". This does make evaluation of some of the complicated limits very easy and is perhaps the main reason for promoting its usage in USA. I think educators in USA want to teach Calculus I (or AP Calculus) in a manner which is just an extension of algebra. Thus the focus in evaluating limits is on usual algebraic operations of $+,-,\times,/$ and $=$ instead of order relations. Students just need to know that if plugging does not work then differentiating and plugging will work. Also such an education emphasizes on "formal differentiation" where students are presented with the rules of differentiation (sum, product, quotient, chain rule etc) together with the differentiation formulas for elementary functions. Such an approach is especially suitable for those students who are taking calculus course from applied perspective ("engineering" students) who are not going to deal with the "rigor" part of calculus anyway.

The above approach is so much contrary to the very spirit of calculus which is essentially based on non-algebraic notion of order relations $<, >$ and completeness. It is essential to grasp the idea that limits are essentially different from value of a function and that they are used to study behavior of values of a function rather than dealing with one value of a function.

Students trained on this approach normally get totally stuck when they encounter $\lim_{x \to 0}x\sin(1/x)$ and that shows the extent of such an approach. Another common problem is that students equate L'Hospital's Rule to "differentiate and plug" and rarely focus on verifying the hypotheses under which it works. Some students don't even bother to check if the expression is an indeterminate form $0/0$ or not.

However there is one plus side of L'Hospital's Rule which I want to highlight. It is easier to teach L'Hospital's Rule as a technique of evaluating limits than teaching the more powerful technique of Taylor's series. The proof of L'Hospital's rule is easier compared to the proof of Taylor's Theorem. Plus the students need to be aware of manipulation of infinite series (multiplying, dividing and composing two infinite series easily at least for few terms) in order to effectively use the technique of Taylor's series.

In my opinion students should be taught all the techniques starting with rules of limits, Squeeze Theorem, standard limit formulas (like $(\sin x)/x \to 1$ as $x \to 0$), L'Hospital's Rule and Taylor's series and preferably in that order.