I’m in AP calc AB, and I’m confused about undefined and DNE, aren’t they both the same? I know the overlap in some instances but there are some problems where it must be either one. I’m confused.
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They are very similar, and the difference can be rather subtle. To help show the difference, consider the function f(x)=sqrt(x). If I asked what is f(-1), the answer is undefined. f(x) has a domain of [0,infinity), but it is not defined for negative inputs. Similarly, f(blue) is undefined, because the function isn’t defined for an input “blue”. However, if I asked what is the solution to f(x)=-1, the answer is DNE. There’s nothing in the definition of f(x) that makes the question illogical, there just isn’t a number whose square root is -1.
Now let’s look at limits. If I ask “what is the limit of f(x) as x approaches a”, what would make this undefined vs DNE? DNE is easy: the function doesn’t approach a limit. For example, the limit of 1/x at x=0 is DNE: it’s negative infinity on the left and positive infinity on the right, so it doesn’t exist. Undefined is trickier, especially without going into epsilon-delta proofs, so I’ll cut to the conclusion: the only way to make a limit undefined is to make one of the inputs nonsense. In other words, the only time a limit is undefined is when the function isn’t actually a function (like f(x)=banana), or when your target “a” isn’t actually a number (like a=purple). If your limit uses an actual math function and you’re approaching an actual number, your limit will always be defined (it just won’t always exist).
Final note: functions can be undefined, and this is probably where a lot of confusion comes from (other than teachers being lazy with vocabulary). The function f(x)=x^2/x is undefined at 0, since f(0)=0/0 and division is undefined when the divisor is 0. However, the LIMIT of f(x) at 0 is defined, and is just 0. Similarly f(x)=1/x is undefined at 0, but the limit is defined and just doesn’t exist. So just remember: functions can be undefined, limits can’t
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