I understand that, for k ≠ 0, lim x-->0 (sinkx)/x is an indeterminate form.
Thus we have to use the fundamental trigonometric limit (i.e. lim x-->0 sinx/x = 1).

Then, my professor asked "What if k = 0?". He said the solution is to use direct substitution and NOT to use the fundamental trigonometric limit. Why is this?

I asked my professor about this and he said it is because sin(0) divided by anything is 0. And a limit is not about finding the functional value??? I don't really understand what he meant. Please help me out.

also, the precise definition of limit, the epsilon delta thingy is out of scope for my course. So, please let me know if I need to learn about that to fully understand what my professor meant. (I'm studying engineering)

I need help with this problem i need to find the area of the big box, the AIs seem to break when i post it and try to give me the answer 1^2 +3^=area of big box=10cm

Which doesn't make sense please i need a clear explanation

I understand that they pulled the 1/5 out of the integral, and I understand that they chose -(x/5) for their u and differentiated it which gives you du = -1/5dx. Next you need to get dx by itself so that you can substitute dx for du and have the whole integral in terms of one variable u right? So to get dx by itself you would multiply both sides by -5 right? Which would give you -5du = dx. You would then swap the dx in the integral for -5du and if you pull the -5 out of the integral as well it get's multiplied by the 1/5 that was already there. -5 times 1/5 = -1, not 25. What am I missing here?

I’ve been thinking. Yk how like after like 7 moves in chess there are 6 sextillion chess positions. So it’s likely that it’s a position never seen before (excluding common long openings) How different is that from words? Like after twelve words is it Likely this exact sentence been said?

(I'd slightly prefer the whole thing to be bilaterally symmetrical, but I don't care that much.)

I'm trying to approximate the shape of an eye using tileing to use for a simple skill tree type thing for perception in an rpg. That's why the number of polygons is so limited.

Also, since the rules say I have to explain the steps I've taken to solve the problem: I've tried using the 3 regular polygons that can be tessellate, but the closest approximation I could get was a rhombus with its ends removed, so I asked my friend who's an undergraduate in math, but he didn't have an answer, so I experimented around with irregular polygons, but I gave up due to how many irregular polygons can be tessellated. So I'm asking people that probably know how to sort through all that for help.

I have a math test tomorrow relating to Pascal’s triangle and factoring expansion. When doing practice, for example, on something like a binomial cubed, I sometimes forget to multiply middle terms by 3 (like 3ab^2). Are there any tips you guys have so I can remember not to make these silly mistakes?

I’m not the brightest with probability problems and I’ve been doing trial and error on my strategy on how to deal with them. I’m in college and some colleagues suggested that I just have to do a lot of practice, which is true but it doesn’t work as well for me compared to studying other math topics. The word play really gets to me leaving me confused and stuck until I just stare at the question. I try to create visual models for every problem I answer but it just fuels my confusion, probably because it doesnt make sense. Sure I get some questions right, here and there which I’ve encountered before, but when I see fresh unfamiliar exam questions, I go blank even though I’ve studied the topics for it. Are there any ways, decision paths, mind maps, structured framework, or approach you guys follow that works well for you whenever you get stuck on a probability problem?

Are the equations x(p+2) = 3 and x = 3/(p+2) equivalent over real numbers if I do not include a domain restriction of p != -2 for the second equation?

They have the same solution set over x, p real at p = -2 both have no solutions, the equations become:

0x = 3, no value of x can solve this

undefined * x = 3, no value of x can solve this

So they are equivalent equations, right? I've asked multiple chatbots this and haven't gotten a straight answer.

Appreciate any thoughts.

Exericse 11.4.10: Use the definition of logarithm to show that (a) log_b(b^x) = x for every real number x, (b) b^{log_b(x)} = x for every positive real number x

Property 7.2.5:

Solution for (a):

Let log_b(b^x) = y
Then b^y = b^x               by definition of logarithm
Then y = x                   by property 7.2.5
Since log_b(b^x) = y and y = x, log_b(b^x) = x, for every x in R
QED

Solution for (b):

Let log_b(x) = y
Then b^y = x                by definition of logarithm
Also, b^{log_b(x)} = b^y    by substitution
Since b^y = x and b^{log_b(x)} = b^y, b^{log_b(x)} = x, for every x in R^+
QED

Is my solution correct?

I am not a true mathematician, so don't be too hard on me for my confusion here.

https://www.youtube.com/watch?v=J72AKZtUpgY

I just watched the first 25 minutes of this video to see how exactly this stuff is taught. It is defined that differentials are changes along a tangent line from a point on a curve. Then it is derived that the slope at the point f'(x) is equal to the ratio of the differentials dy/dx.

Question 1: If we remind ourselves that dy/dx as a ratio of differentials is not the definition of the derivative, but is rather "derived" to be equal, is there really a problem here? I always hear it is non-rigorous, but particularly when looking at it in this way I don't see a problem yet.

Question 2: The only issue is then An issue I see is dy/dx is being used as notation for the ratio of two finite numbers, and simultaneously as the limit of a ratio (how derivative is defined in real analysis). So we have f'(x) (derivative) = dy/dx (derivative) = dy/dx (ratio of two numbers). Now this is problematic. But to me it seems more like an abuse of notation than lacking rigor.

Can we really hammer down where the non-rigor lies? I'd really appreciate if possible people could respond in line to the two questions. I am comfortable with real analysis perspectives, where differentials are never mentioned. But I look back on my calculus days, and have a hard time pinpointing the "problem"

Putted in Desmos so its easier to show lol (Typed out each step I did on my paper)

So I started by removing absolute value (since both are positive)

Went ahead and set up 2 equations (One Positive 8, One Negative 8)

Subtracted 5 from both equations (giving me 3 and -13)

Divided 2 from both equations (giving me 1.5 and -6.5)

used interval notation (Neg Infinity, -6.5] U [1.5, Pos Infinity)

anything that I'm potentially missing?

Tried doing this problem but it doesn’t make sense to me. I’ve tried making the bigger rectangle equal 1000 but that’s not right. Don’t know what it means by what does it equal.

Hi Everyone, I recently posted in this reddit about my proof for a Twin Prime Conjecture.

In that thread I got feedback that my Theorem 3 and Lemma 5 were broken and I had gaps in my logic.

I believe I have addressed those issues now, and so if you are willing I am grateful if you can have a second look.

I am looking for people to show me where the proof is currently weak as they see it.

Because, I think I have it, and it's rigorous, and if I don't have it, I think the remaining gaps in logic are small now if there are any at all.

I do intend over the next week to attempt to put this into Lean and try to verify it that way.
But in the meantime, I am here to ask for some potential roadblocks any of you might see for me doing that, so I can catch them earlier, because I am unfamiliar with Lean.

This new paper fixes the problems brought up about my previous Theorem 3 and Lemma 5.

It is the same mathematical construction, however I have approached the logical statements in a different method than I previously did, and I believe I have overcome the hurdles now.

This paper also has a companion python function which works as a finite representation of how the function works.

The python function itself cannot act as the proof because you would have to run the function for an infinite length of time with infinite computer resources in order to use that as proof.
However it does serve as an easy to understand implementation of the function and proof posed in the paper.

A quick read of the Abstract can give you a good understanding of exactly how I am intending to solve the TPC

Thank you in advance for anyone willing to spend some time on this to help me out.
Also I am hosting the paper on my own site incase Zenodo is down.
https://prosperousplanet.ca/research/TwinPrimes.pdf

The inverse of xe^x is the Lambert W function, whereas the inverse of x^x is called ssrt(x) in wiki I think. They can also be expressed in terms of each other.

I am wondering why the Lambert W function is more popular than the inverse of x^x, when tetration is the 4th hyperoperations(+,x, exponentiation, then tetration). Since we also study the inverses of addition, multiplication, exponentiation, but why not the inverse of tetration?

According to the table, can you infer about f(x)?

The options were lim_{x→2} f(x) = 1, lim_{x→2} f(x) = ∞, lim_{x→10} f(x) = ∞, lim_{x→10} f(x) = 2.

Obviously, the given table matches the function f(x)=0.75-0.05*log10(|2-x|). One of my friends had the same thought process, so they picked the second choice. I knew that the teacher wanted us to choose the first one, so I did.

According to the teacher, the correct answer was the first one, but I feel that it isn't fair that my friend didn't get points. We tried to argue, but she wouldn't give the points.

Is it my friend's problem or the teacher's problem that he lost points on that problem?

This is a question about whether there is a function that would show all 3 of these given changes in variables. I mean the shortest function possible (even if its necessarily large).

In tasks where you're looking to disprove isomorphisms, you usually start from group order or Abelian property etc. and work your way down the rabbit hole. One of the most crucial 'tests' is counting the elements of certain order. This, however, can be quite daunting and I don't fully understand it, especially when it comes to non-cyclic groups like dihedral, symmetric and alternating.

I learned one method that I will try explaining here, but it works only Z. So let's say we have Z56 and we want to calculate the number of elements of order 8 (56 is divisible by 8, hence that certainly is a legitimate order for an element).

The first thing I would do is factor 8 into coprimes, so I need 2^3. Now, I write out Z56 in elementary form: Z8 x Z7, and there we have it. So, we have exactly one 2^3 and we need 2^3 elements from it.

Setup: k = 3, because that's the degree we need. 2^min{n, k}, where n is the power of the cotresponding group in the elementary form and k is the power of the wanted element.

So, 2^min{3, 3} = 2^3 = 8. But then I do k - 1 right after so 2^min{2, 3} = 2^2 = 4 and finally 8 - 4 = 4. The correct number is 4, there are 4 elements of order 8 in Z56.

However, this specific trick only works with Z (and potentially Euler group Z/nZ*). My question is how to perform similar things for dihedral, symmetric and aletrnating groups. Dihedral ones are quite easy because they're usually small and can be calculated fast, but I am completely confused around symmetric and alternating groups as I don't understand them fully.

I am joining the union and this is a sample question and I am just not sure how to figure this out. The arc is throwing me off, it's the only one I am having trouble with. If someone could explain this to me I would be extremely appreciative. Thank you so much.

The game is Pokémon, protect (and spiky shield) have a 2/3 chance of failing when used twice (which is the case) and blizzard has a 70% chance of hitting.
What is the chance of neither Pokémon protecting and both getting hit by blizzard?

My wife and I did a fantasy football draft last night and we determined the draft order with a card game. There were 12 players, and every turn we would all draw cards until one person drew the one “bomb card.” Whoever drew that card was “out” and drafting at that position. (Ex: the person that drew the bomb card in round 1 drafted 12th. The person that drew the bomb card in round 2 drafted 11th, and so on) We would remove one of the safe cards, and draw until one person was left.

Afterward, my wife and I were talking about the game and I commented how odd it was that it was coming down to the last one or two people every turn. And she — who is incredibly smart and fantastic with numbers and works with them all day — said that it actually makes sense because the people drawing first are more likely to draw a safe card.

But I argued that they are more likely to have someone ahead of them draw the bomb card and that it feels like the people drawing in the middle of the group — who have neither benefit of having the best odds of drawing a safe cards nor the highest amount of people drawing ahead of them — have the highest chance of drawing the bomb card.

My wife is so much smarter than me in just about every way I can imagine and certainly with math problems, so it wouldn’t surprise me at all to be told that I am wrong here… but am I?

Edit to answer some of the questions that have been asked: I shuffled the deck before every round, and spread out all the cards so that each person could which ever one they wanted.
The back of all the cards looked the same.
Whoever drew first would pick the card, show the group what card it was, and then the next person would do the same.

I have this task and I don't know how to solve it despite having all the basic knowledge required for it. I can't 'connect the dots' so to speak.

a) Find the max element order in S_{10} - symmetric group consisting of 10 'base' elements and its order being 10!

b) Let s be some element of max order from S_{10}, find sgn(s) as well as s^{2040}.

c) If H is a subgroup of S_{10} and H contains an element of max order, then H contains the equal number of even and odd permutations. Prove it.

I've never encountered a task like this so I genuinely don't know how to think it through.

a) My intuition says I should find the lcm of prime cofactors of 10, so lcm(2, 5) = 10. I genuinely don't know how to do it. This also doesn't add up in my logic because 2+5 = 7. So maybe lcm(2, 3, 5) = 30? But I arrive to that answer by pure luck and need some explanation why is that.

b) Would depend on a) but let's say it is 30. What is sgn(s)? I've never seen it before. I do recall it might be odd vs even number of elements in a single disjoint cycle, odd ones starting with a minus. But how would I take that order of 30 and split it into smaller, disjoint cycles? so I would have sgn(s2) sgn(s3) sgn(s5) where s2, s3 and s5 only represent the disjoint cycles of 2, 3 and 5 elements. So in front of sgn(s2) there will be a + and in front of 2 others - so it would be positive in the end?

s^{2040} I did similar stuff before, I can do 30 = 0 mod(2040) (because 30*70 = 2100, 2100-(2*30) = 2040) so therefore s^{2040} = s^0 so nothing changes (and even if it did, I have no given s explicityl so I have no idea what am I supposed to do, s^{2040} only shifts all the elements in each disjoint cycle, but here s isn't given so there are no specific disjoint cycles i have to shift, confusing)

c) Most challenging part and the one I don't understand at all. Need full help and ELI5 explanation for this one haha. Whenever there's something to prove like subgroups, ideals, normal subgroups etc. I am completely lost even though I know I only need to show non-emptiness, closure under operation and closure under inverses. I just don't know how...

If you did read everything, thanks a lot!!!

My son just started High School algebra in 8th grade. His teacher gave him homework on which he said each equation needs a 1 step solution. My son asked me for help on the equation in the attached picture, I helped him get the answer but in 4 steps and I cannot figure out for the life of me how to simplify this to 1 step. Please help:

5r-5-6r=-6
Adding 5 to both sides becomes 5r-6r=-1
-r=-1
r=1

to be clear im not saying im some genius or anything i've just truthfully never seen this shape before and i wonder if it can be mathematically replicated and turned into a fractal. if it exists what is it's name? (sorry for bad drawing)

UNderstanding math vs:

Things like:

• Misreading the question
• Copying a number incorrectly
• Making a calculation error
• Forgetting a step
• Using the wrong formula
• Getting an answer that looks reasonable but is actually wrong
• Rushing because I think I know how to do it

It’s frustrating because sometimes I understand the concept perfectly, but still lose marks because of one stupid mistake.

For those of you who are good at maths, what habits or checking methods do you use to reduce mistakes?

I’d especially appreciate advice that actually works rather than just “pay more attention”.

Thanks!

Set Operations:

This shirt makes ZERO freaking sense. Basically set of all elements that aren't in b or c and thats what I freaking did. Istg math is so freaking stupid.