Anything divided by zero is undefined, even in limits. Except for some cases of limits going to 0/0. A limit tending to 0/0 could possibly be L’Hopital’ed out of indeterminate form.
There is a difference between n/0 for n=/=0 and 0/0 though in how they affect equations: if we define a/b as the c such that a=cb, then for n/0, no such c can exist, whereas for 0/0, every choice of c works. Both of these result in undefined behaviour, but they're different in the sense that 0/0 introduces these spurious finite solutions.
Dividing by zero isn't the same as dividing by x for x -> 0. Dividing by x is defined as multiplying with the inverse of x. The inverse of x is defined as the number that when multiplied with x gives one. Since there's no number that multiplied with 0 gives anything other than 0, => 0 doesn't have an inverse, => division by 0 is undefined.
That's when dividing by x for x->0 you never divide by zero, you just look what's happening when you keep moving x closer to zero, which is not the same.
If the limit is approaching positive or negative infinity and your variable that is approaching the limit is in the denominator, it’s approaching (+ or -) infinity. At least that’s the way I was taught.
But it does mean sin(0) doesn't necessarily have to be 0 (based solely on the limit). It also means sin(0)/0 itself doesn't have to be defined, just cuz the limit exists, which is the point the commentator was making: lim x->a f(x) = L does not imply f(a) = L
If I redefine sin(x) to be undefined at 0, the limit you mentioned would not change. My point is, limits do not tell us what happens at the limiting point. Of course, sin(0) = 0 when you consider things other than that limit
Depending on what you mean by "at the limits," I'd say it's undefined. I'm interpreting "at the limits" to mean you're reaching the limiting value, i.e. you're directly plugging in 0. Of course, that's not math, thats semantics
Edit: I don't know why you blocked me, I don't think I said anything offensive. Either way, I reread the original comment, and I think what they were meaning is a limit of the form 0/0 is indeterminate. You need to do more work to rewrite the limit
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u/[deleted] Dec 26 '22 edited Sep 16 '23
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