99% Get This WRONG. It's Not Simple. 🧠

[@RyanLewis-Johnson-wq6xs]

X^2/X=If x isn’t zero

[@mallusrgreat]

Simplifying has failed us

[@dawood1106]

It felt like you were yapping but you were on to something

[@NumberNinjaDave]

@@dawood1106 what did you learn from the video

[@HoSza1]

​@@NumberNinjaDaveit gets even worse once you examine what would happen when you allow x to be complex. x²/x -> i when x -> 0 "from imaginaries above" and x²/x -> -i when x -> 0 "from imaginaries below" (meaning x=0+bi and x=0-bi respectively such that b->0 and b>0 for both cases).

[@NumberNinjaDave]

@@HoSza1 very true! The normal cartesian plane is just a projection of the complex plane where the ordinate is 0

[@HoSza1]

@@NumberNinjaDave I should take a look at some visualisations of the functions z->z and z->z²/z and compare them to understand what kind of discontinuity does the latter have at (or near) z=0, though since these are basically 4D graphs (in the same sense that graphs of R->R functions are 2D, drawn on the Cartesian plane), they are too hard for me to grasp them.

[@NumberNinjaDave]

@@HoSza1 that makes sense. I’m not a visual person myself in 3d or 4d space so those do get tricky

[@Dissimulate]

If the limit exists, I integrate over holes like a bulldozer.

[@NumberNinjaDave]

@@Dissimulate sweep the leg, too

[@RyanLewis-Johnson-wq6xs]

That’s why we use limits lim((x^2)/x) when x approaches 0 it equals 0

[@RyanLewis-Johnson-wq6xs]

0/0=indeterminate It has no exact value.

[@NumberNinjaDave]

@@RyanLewis-Johnson-wq6xs Very true, ninja

[@jkid1134]

for every value in the domain of x²/x, x has the same value. that is to say, x extends x²/x to R. presenting the lesson here as "these are different; there's a hole" instead of "yes, over R\0" feels weird, when the similarities between functional extensions are almost always the important part. I for one think of ln(x) and Log(x) as "the same function, but Log(x) can take complex arguments", and I would be surprised if you don't.

[@NumberNinjaDave]

@@jkid1134 good perspective, though I see ln and log as different due to ln being a subset of the space of the log function where it intersects the base as e.

[@jkid1134]

@@NumberNinjaDave lest there be any confusion here, Log(x) here is the principal value of the complex logarithm. maybe the least clear example I could have picked.

[@emad3241]

These kind of things are very important in rigorous mathematics, when you perform any mathematical operation on an equation, you must respect the number space that the set of equations you produced satisfies for example x=x is only true when x!=0 At least when x satisfies x^2/x=x this is very important to avoid false proofs

[@NumberNinjaDave]

@@emad3241 bravo, ninja

[@BARB3LLA]

The answer is really simple. Let's say f(x)=x and g(x)=x²/x, for f(x) to be Equal to g(x), 2 things must be true. Df=Dg and f(x)=f(g). In this case, Df=R and Dg=R* so since Df≠Dg f(x)≠g(x). Really liked the video btw, you gave some Very good explanations👍🏻👍🏻

[@NumberNinjaDave]

Thank you! And great explanation

[@dhavamaneeganesh2147]

I thought that all people who are bit decent at math know this? What's your take?

[@NumberNinjaDave]

That's subjective. Not everyone is the greatest at math. I'm hoping to help with that

[@baconboyxy]

@@NumberNinjaDave While I appreciate the content, the clickbaity “99% get this wrong” title is kinda lame. Like Americans are dumb, but not 99% don’t understand divide by 0 dumb.

[@matei_woold_wewu]

x²/x=xx/x=x

[@NumberNinjaDave]

What if X is zero

[@TheVergile]

…whats the point of the limit then? even if the function appears continuous and has a limit it still fails at zero. which was already readily apparent from the very beginning. there is no scenario in which dividing by zero would be okay, regardless of the continuity and limit. no idea why you are saying its not that simple when apparently, yes, it IS just that simple. dont divide by 0.

[@NumberNinjaDave]

Not every student understands the concepts and that’s why they come to this video

[@carultch]

@@NumberNinjaDaveAre there any practical applications of functions with removable discontinuities, where it isn't enough to just take the limit of it, and remove the discontinuity? One apparent application I can think of, where holes come up in such a function, is a pole coinciding with a zero in the Laplace domain, for control system theory. However, holes in these functions don't end up governing the time domain response anyway, and can be removed as if they weren't there in the first place.

[@jackkalver4644]

It’s also indeterminate for infinite values and for infinitesimal values.

[@NumberNinjaDave]

Not necessarily. For very small epsilon, so long as you know if epsilon is slightly greater than 0 or less, you know which side of the origin you are on, and you'll know if this diverges to positive or negative infinity. Futher, as x approaches positive infinity, we can safely say that x^2 / x represents a non zero x, and that the limit diverges to +inf, or -inf if to the left of the origin. Divergence at infinity and indetermination are two different things due to the latter having ambiguity due to discontinuity.

[@user-lu6yg3vk9z]

This a good test question.

[@NumberNinjaDave]

@@user-lu6yg3vk9z for sure

[@user-lu6yg3vk9z]

@@NumberNinjaDave basically u multiple by a form of 1 x/x which created a removable hole which therefore lead to a change in the domain.

[@NumberNinjaDave]

@@user-lu6yg3vk9z you’re a 🥷

[@marvhollingworth663]

Why is 0/0 not = 1? You're dividing it by itself.

[@NumberNinjaDave]

Great question. Division by 0 is simply undefined. How can you fit a value into nothing?

[@NumberNinjaDave]

@@marvhollingworth663 as a visual, graph y =1/x. Approach the origin from the left. Now do it from the right. There’s ambiguity on the converging value

[@adiaphoros6842]

​​​@@NumberNinjaDave but since x²/x uses the same number on the numerator and denominator, it should be 1. This makes 0 a removable discontinuity. If the expression was y²/x, then I would understand why lim (x,y)→(0,0) is undefined.

[@tingbrian5437]

lets start with the concept of divison. 12÷3 meaning that you have a total amount of 12 apples, and you are distrubuting apples 3 by 3, and you are finding the number of people that can get the apple, which is 4(12÷3=4). however, for 12÷0 meaning that for a total of 12 apples, you give out 0 apples to people and finding the number of people you can give. The answer will be infinity as you are not giving any apple to someone, such that you can give apples to infinite amount of people while you still have 12 apples on your hand. Does it solve your question?

[@NumberNinjaDave]

@@tingbrian5437 except the answer isn’t infinity. It’s undefined.