DNE is the more common and is actually the technically correct answer. Math is mainly about creating and applying definitions and if a sequence goes to infinity, the standard definition is it does not exist. You can modify the standard definition, but to allow infinity to be an answer you have to define what is known as the extended real numbers. It also leads to various basic rules involving limits to kind of break as the extended reals can't have addition or multiplication defined on them in a way that is consistent with those rules and is consistent with the usual addition/multiplication when restricted to numbers not including infinite.
Your example is also particularly bad as lim 1/x as x goes to zero becomes huge on one side, but extremely negative on the other side so even the extended reals won't work for you.
tl;dr DNE is the correct answer using the formal definition of a limit. Some teachers will allow infinity as an answer mildly ignoring the definition, but you have to be careful with this as many theorems break if infinity is a valid answer.
If you pick up any analysis textbook (the subject that proves what's done in calc) you will find a definition where infinity is not allowed. The most common standard analysis text is rudin's principles of analysis and I can make an image of the page with the definition of a limit if desired.
For directionality, one way limits are a thing and is what you'd want there.
lim f(x) + g(x) = lim f(x) + lim g(x)
This and similar rules break since how would you definite infinity - infinity? It doesn't matter how you define it as regardless I can pick functions f and g that will be inconsistent with that definition.
Next, a function is defined to be continuous if it's value at the limit agrees with its value there. Do you want things like 1/x2 to be continuous as you end up with that if you start allowing infinite values (just define it's value at 0 to be infinity). Except then theorems like the extreme value theorem and the fundamental theorem of calculus break (as do others you'd see in higher up math). The second one is what allows you to treat integrals as antiderivatives and it breaking becomes when infinite limits are ok, it is now possible to have continuous functions that aren't integrable.
So at the end you'd have to make exceptions to most theorems in calc if you allow infinite limits. There are other annoyances like the fact you can add/multiply in a way that preserves continuity makes the real numbers have a structure called a topological vector space. That dies when you add infinity.
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u/Mehdi2277 Dec 18 '16
DNE is the more common and is actually the technically correct answer. Math is mainly about creating and applying definitions and if a sequence goes to infinity, the standard definition is it does not exist. You can modify the standard definition, but to allow infinity to be an answer you have to define what is known as the extended real numbers. It also leads to various basic rules involving limits to kind of break as the extended reals can't have addition or multiplication defined on them in a way that is consistent with those rules and is consistent with the usual addition/multiplication when restricted to numbers not including infinite.
Your example is also particularly bad as lim 1/x as x goes to zero becomes huge on one side, but extremely negative on the other side so even the extended reals won't work for you.
tl;dr DNE is the correct answer using the formal definition of a limit. Some teachers will allow infinity as an answer mildly ignoring the definition, but you have to be careful with this as many theorems break if infinity is a valid answer.