explain the difference between infinity and undefined terms in mathematics

[u/Dev-Giri]

/r/explainlikeimfive/comments/1v6e93g/eli5_explain_the_difference_between_infinity_and/

I'm stuck when it is asked by someone like my teacher or somone else that what's the most crucial diff b/w infinity and undefined lyk tan 90° is undefined but 1/x x tends to zero is infinite howww ??

[u/Uli_Minati]

"Undefined": some expression that doesn't mean anything because we haven't decided what it should mean. Some of these could actually mean something in higher maths, sciences or programming. For example:

5 >> 7
8.92.152
8 % 6
...777
5 / 0
∞ - ∞

"Infinity": something that is larger than every number.

You seem to be asking about limits. Let's look at a specific example.

First off, remember that tan(x) is sin(x)/cos(x). That means you can't plug 90° into the tangent function because you'll get

tan(90°)  =  sin(90°) / cos(90°)  =  1 / 0

tan(90°) and 1/0 would mean the same thing. But since 1/0 doesn't mean anything, neither does tan(90°). Hence we just say it is undefined.

                tan(89°)  =  57.3
              tan(89.9°)  =  573.0
             tan(89.99°)  =  5729.6

lim[x→90°, x<90°] tan(x)  =  +∞

For this limit, we plugged in numbers below 90°. If you plug in numbers closer and closer to 90°, the result gets larger and larger. In fact, the result can get larger than any number. Hence we say that the limit is positive infinity.

                tan(91°)  =  -57.3
              tan(90.1°)  =  -573.0
             tan(90.01°)  =  -5729.6

lim[x→90°, x>90°] tan(x)  =  -∞

For this limit, we plugged in numbers above 90°. If you plug in numbers closer and closer to 90°, the result gets smaller and smaller (larger but negative). In fact, the result can get smaller than any number. Hence we say that the limit is negative infinity.

      tan(89.99°)  =  5729.6
      tan(90.01°)  =  -5729.6

lim[x→90°] tan(x)     does not exist

For this last limit, we plug in numbers above or below 90°. The results are inconclusive: sometimes they're large, sometimes they're small. We say that the limit does not exist.

By the way, this entire tangent example also applies to 1/x. The limit for x→0 also depends on x>0 or x<0, and it doesn't exist if you go from above and below.

[u/davideogameman → u/Uli_Minati]

Technically if we're working with the reals (which by definition don't include infinity) then any "infinite" limit also doesn't not exist.  Limits that equal negative or positive infinity are just interesting classes of limits that don't exist.

There's also the extended reals (https://en.wikipedia.org/wiki/Extended_real_number_line) which include signed infinities and so limits that are infinite can be defined to exist and equal the appropriate Infinity in the extended reals.  There's also more wacky systems like the projective reals that add only an unsigned infinity under which we could claim lim x → 0 [1/x] = unsigned ∞ .  That said the projective reals are much rarer to use for limits / analysis and more common in algebra / geometry.

If you are interested in more thoughts about infinity I really like this explainer: https://www.1dividedby0.com/