I teach mostly physics, but have taught calculus a couple of times. From the physics end, I see things in almost the opposite way that you do. Here is a typical way that this plays out in my class.
We have a homework problem where a cable is stretched between two buildings, with a streetlight hanging from the middle, so that the cable makes the shape of a "V". The students calculate a formula for the tension $T$ in the table as a function of the height $h$ by which it sags in the middle. The problem asks them to do some interpretation of the result, including checking its units, the trend of $T$ as a function of $h$, and the special case of $h=0$. (Note that we automatically have $h\ge 0$ in this problem.)
A student comes to my office hours for help with the problem, and we get to the point where we're checking the special case $h=0$. They say, "Oh, it's undefined, because zero is in the denominator."
At this point they think they're done, and they need to be disabused of this notion that has been inculcated in them. They are missing out on the insight that is to be gained by realizing that the result is infinite.
If you like, you can dress this up in various language such as the language of limits, or saying that the result is undefined as a real number, but only because the result is infinite, and we don't have infinite real numbers. But these are issues of mathematical formalism that are way above the level of this type of student. To see the actual level that they're operating at, it may be useful to consider the following fragment of dialog, which I have had many times with many students:
Student: "Oh, it's undefined."
Me: "That's great that you've had such good mathematical training, and your math teachers have told you that dividing by zero is undefined. But suppose we really did divide by zero. What do you think you would really get?"
Me: "Well, hang on, what kind of result would you get if you divided by a really small number, like 1/0.001?"
Student: "It would come out really small?"
Me: "Let's try that on your calculator."
Student: "Oh, it's big. I see."
In other words, they're not at the level where they understand that the expression diverges but they just don't know how to formalize that statement. They're at the level where they lack the idea that it blows up at all. They need a first exposure to the idea that it blows up, and then on some later pass maybe they can learn about calculus, limits, the extended reals, and so on. The first exposure needs to happen in grade school, when they first learn about division. If they learn division at age 9, and calculus (possibly) at age 18, then IMO the right answer for them to give from age 9 to age 18 is that $1/0=\infty$ (or possibly $\pm\infty$, depending on context).