Pretty nice video. As a math student I have to comment. What you've been describing, with things like "x divided by a value like 0.0001 is x * 10^y" is an intuitive way to arrive at the definition of a limit. It is indeed true that adding more and more zeroes before the digit 1 gets you closer and closer to 0; however, you never get to exactly 0. It is also true that 1 divided by this number gets bigger and bigger, and is exactly equal to 10^y. Because this value gets bigger and bigger the smaller x gets, we say that the limit of the function from 1/x as x goes to 0 from the positive side is "equal to" infinity. The first problem is, is that 0 is not a (real) number, so saying something we want to be a number to be "equal to" infinity means that that something isn't rally a number. In mathematics we say that this value diverges to infinity, meaning that it gets larger and larger the closer we get to 0. The second problem is that this ignores negative numbers; those are numbers smaller than 0. For example, there exists a number called "negative one" (-1), which we can add to the number one to get the result of 1 + (-1) = 0. Instead of dividing by a number that is close to zero but greater than 0, we can also divide by a number that is close to 0 but smaller than 0; in this case the result we will get is that 1 / -0.00001 = - 10^y, where y is again the number of zeroes. The closer we get to 0, the bigger y gets, and the more negative - 10^y becomes. Therefore we say that, in a similar fashion, 1/ z diverges to -infinity as z goes to 0 from the negative side, because it decreases without bound. Now for the true limit to exist, both of these limits must exist and equal the same value - which they sadly don't. Hence, 1/x has no limit for x approaching 0. Your second idea of setting 0/0 = 1 is also an application of a limit. You noted that (some value) divided by (the same value) is always 1, but this actually does not hold if that value is 0 (because of division by zero). You did correctly point out that for really small positive numbers, the value remains stable at 1. This holds for any value close enough to 0 but not equal to 0, even negative ones - and hence we can say that the limit of x / x as x goes to 0 is equal to 1 (written as lim_{x -> 0} (x / x) = 1, where _{Text} denotes that the Text is written as small letters below "lim"). This however does not mean that x / x is equal to 1 when evaluated at 0, and certainly not that 0 / 0 = 1. Setting 0 / 0 = 1 leads to varies problems, mostly related to the nice rules we want division to follow. For one, we would like that a * b / c is the same when we first multiply a and b and then divide that by c, as well as first dividing b by c and then multiplying a by that fraction. However, this rule does not hold anymore: (2 * 0) / 0 equals 0 / 0, thus 1, but 2 * (0 / 0) = 2 * 1 = 2. That would mean that 1 = 2, and we certainly don't want that! This problem arises because 0 / 0 is not only the result of letting x = 0 in x / x, but also the result of x = 0 in (2 * x) / x. The latter is always equal to 2 for any x besides 0, and has a limit of 2 when x approaches 0. Lastly, the two results of division by 0 conflict. On the one hand, x/0 = infinity for all x. But on the other, 0/0 = 1. But what if we set x = 0 in x/0? Is this both infinity and one simultaneously? In summary, a neat way to stumble upon limits. But be careful when you go from a limit to concluding that something actually is that certain value. Let me know if you have any questions/
Pretty nice video. As a math student I have to comment.
What you've been describing, with things like "x divided by a value like 0.0001 is x * 10^y" is an intuitive way to arrive at the definition of a limit. It is indeed true that adding more and more zeroes before the digit 1 gets you closer and closer to 0; however, you never get to exactly 0. It is also true that 1 divided by this number gets bigger and bigger, and is exactly equal to 10^y. Because this value gets bigger and bigger the smaller x gets, we say that the limit of the function from 1/x as x goes to 0 from the positive side is "equal to" infinity.
The first problem is, is that 0 is not a (real) number, so saying something we want to be a number to be "equal to" infinity means that that something isn't rally a number. In mathematics we say that this value diverges to infinity, meaning that it gets larger and larger the closer we get to 0.
The second problem is that this ignores negative numbers; those are numbers smaller than 0. For example, there exists a number called "negative one" (-1), which we can add to the number one to get the result of 1 + (-1) = 0. Instead of dividing by a number that is close to zero but greater than 0, we can also divide by a number that is close to 0 but smaller than 0; in this case the result we will get is that 1 / -0.00001 = - 10^y, where y is again the number of zeroes. The closer we get to 0, the bigger y gets, and the more negative - 10^y becomes. Therefore we say that, in a similar fashion, 1/ z diverges to -infinity as z goes to 0 from the negative side, because it decreases without bound.
Now for the true limit to exist, both of these limits must exist and equal the same value - which they sadly don't. Hence, 1/x has no limit for x approaching 0.
Your second idea of setting 0/0 = 1 is also an application of a limit. You noted that (some value) divided by (the same value) is always 1, but this actually does not hold if that value is 0 (because of division by zero). You did correctly point out that for really small positive numbers, the value remains stable at 1. This holds for any value close enough to 0 but not equal to 0, even negative ones - and hence we can say that the limit of x / x as x goes to 0 is equal to 1 (written as lim_{x -> 0} (x / x) = 1, where _{Text} denotes that the Text is written as small letters below "lim"). This however does not mean that x / x is equal to 1 when evaluated at 0, and certainly not that 0 / 0 = 1.
Setting 0 / 0 = 1 leads to varies problems, mostly related to the nice rules we want division to follow. For one, we would like that a * b / c is the same when we first multiply a and b and then divide that by c, as well as first dividing b by c and then multiplying a by that fraction. However, this rule does not hold anymore: (2 * 0) / 0 equals 0 / 0, thus 1, but 2 * (0 / 0) = 2 * 1 = 2. That would mean that 1 = 2, and we certainly don't want that!
This problem arises because 0 / 0 is not only the result of letting x = 0 in x / x, but also the result of x = 0 in (2 * x) / x. The latter is always equal to 2 for any x besides 0, and has a limit of 2 when x approaches 0.
Lastly, the two results of division by 0 conflict. On the one hand, x/0 = infinity for all x. But on the other, 0/0 = 1. But what if we set x = 0 in x/0? Is this both infinity and one simultaneously?
In summary, a neat way to stumble upon limits. But be careful when you go from a limit to concluding that something actually is that certain value.
Let me know if you have any questions/
I aint reading all that 🤗
Bro you should be a english major not a math student wtf