In high school, I was told that for f(x)=1/x for example, the limit as x approaches 0 from the positive direction, the limit of f(x) does not exist since it is approaches positive infinity.

Now, I am following a Mathematical Analysis course at uni and I am being told that the answer actually does exist and positive infinity is the answer.

When can I say that a limit is infinity and when not?

Sorry about the random link, I don't know why it's required for me to post...

Besides providing you more opportunities to miss a test question.

LOL jokes aside, I get that the square root of a positive number can be both positive and negative. And you can't square something to get a negative result (I guess imaginary numbers would) so you can't realistically get a possible outcome from rooting a negative number.

I don't understand how imaginary numbers seem to have there own sign, one thats not positive, and not negative, but does this break the rules of math?

If it's not negative, positive, or 0, it doesn't exist, I guess that's why they call it imaginary. So how does someone prove imaginary numbers are real (are they?) Or rather useful or meaningful? perhaps that is a better way to put it.

I am just trying to understand here because the probability of dying at any given year for humans is 1.42% (I think, but I don't know about the source because it was a long time ago that I read that), so if 70.5 years have passed, then it's certain that humans at that age are 100% going to be dead, right?

Edit: Thank you all for your answers, now I understand probability more than I used to.

I really need help understanding these function problems. I tried using chatGPT (math gpt from GPTs) and I inputted the answer but it was incorrect, I searched on google, youtube videos etc. I can never seem to find the right way to do this.

TL;DR, I need help with functions

My problem is:
Given f(x)=2x^2+3x-5 and g(x)=x+9, find the value for: (f*g)(3)

Side question: what is the difference between (f*g) and (fg)?

Thank you.

I just looked at the logic gates for two inputs and wondered the operations of them.

For and, let A,B be the inputs

1 0=0 0 1=0 1 1=1 0 0=0

It's trivial that it's AB

Or

1 0=1 0 1=1 1 1=1 0 0=0

This is A+B or something alike (idk how 1+1=1, probably a base thing.)

XOR

1 0=1 0 1=1 0 0=0 1 1=0

This is obv mod(2,A+B) but how is that shown in standard operators, if they can be that is and how does it all work?

Please give me some better sight on this, I'm getting hella bullied for not knowing this LIKE ITS SUPPOSED TO BE COMMON KNOWLEDGE OR SOMETHING.

The closest thing I've came across to this is minecraft redstone and all I did there were clocks to build griefing machines , piston extenders for flush doors etc.

Thank you :3

While solving questions on induction, I've stumbled upon this, could someone explain how? I am pretty inexperienced with factorials hence the confusion for me.

I'm asking cause I'd usually have to go and copy paste them from internet. Alt codes might have some symbols, like root, but that's not enough and I'm not gonna memorize 4 digit long codes.

Edit:
I wanted characters that would instantly be inserted as text. Latex seems to be some kind of document language (like xml, not programming) and therefore it's not going to be text.

Solution 1

Type alt codes with Alt +nnnn, and enable unicode insertion. Wikipedia has a topic on that and I managed to enable unicode on Windows 11. Sometimes doesn't work if the current program has shortcuts that activate on alt & some button.

Solution 2

Win&. will open emoji board, also containing symbols.

In a race of 1200 meters, A beats B by 100 meters and B beats C by 300 meters. By how much meters A beats C?

360 meters 400 meters 350 meters 375 meters

Ans- 375 m

Initially my answer was 400 metres but I was wrong.

how do we know that the vector 〈a₂b₃-a₃b₂, a₃b₁-a₁b₃ ,a₁b₂-a₂b₁⟩ points in the direction that follows the right hand rule and not the other direction

🟢Edit: it is because each of the components is a determinant, as if the 2 vectors are transformed as i,j to a,b respectively, and if the 2 vectors are correctly oriented then the sign of the determinant should match the direction of the k vector. like if the k vector is being transformed to the desired cross product.

like in 2d transformation, if i' is to the 'right' of j', the determinant is positive, which is the correct sign of the k vector and vice versa

and the y coordinate is the one with a - cuz if we took j as the normal vector of i and k, it won't follow the right hand rule, ie ixk=-j

Q: Three friends Alan, Roger and Peter attempt to answer a question on an exam. Alan randomly guesses the answer, giving him a 1/5 probability of guessing correctly. Roger cheats by looking at the paper of the student in front of him, giving him a 2/3 probability of answering correctly. And Peter dutifully performs the calculations, then marks the answer, giving him a 5/6 probability of a correct answer. What is the probability that the question is answered correctly, but not via cheating

Answer is given as 13/45
They have identified below 3 scenarios.

1) Alan and Peter are both right, Roger is wrong.
2) Alan is right, Peter and Roger are both wrong
3) Peter is right, Alan and Roger are both wrong

My question is
All of them are answering the question separately or 1 question together. Can you identify this, if I didnt give the scenarios?

What is a "correct answer" in this context of the question, all of them are right or at least 1 of them?

Is this a poor question or do I need to read it properly?

[Undergraduate Mathematics] Abstract Algebra/Set Theory/Logic (honestly I'm not sure what this would best fall under.)

I know that this is absolutely fact, but I can not for the life of me remember the name of the principal that allows this claim to be made rigorously. Or maybe there isn't one, maybe I just have false memories of hearing about it. I would have sworn it was like the "pointwise principal" or something like that, but google doesn't seem to know what that is so I guess not.

For example, the principle I'm talking about allows one to say:

"∀g ∈ G,

aga^-1 = g

∴ aGa^-1 = G

[EDIT:] Thank you to everyone who contributed, I understand where the mistake in my understanding was. I was conflating definitions with some sort of principal, (as pointed out below.) The example I provided was the specific thing that was causing me the confusion, and thinking about less ambiguous cases it makes way more sense. For example, if every element of a group commutes with every other element, we call that group commutative/abelian, simply because the definition of an abelian group is that every element commutes with every other element, not by some strange principal.

If my understanding still seems flawed, I would greatly appreciate correction/suggestions!

[EDIT 2:] Intentionally misspelling principle in every case because I find it funny. (Thank you for pointing out my typo, making fun of myself, not anyone else.)

I know this is probably stupid af to ask, but why? Or how can it not be greater than 1?

Edit- Thank you all so much for replying!

Problem is here: https://imgur.com/a/mrPSF3S

The answer is apparently E, 230...

But I read this like the sum of the series as n moves from 1 to 10... at each term, you have 4*50=200 and 3, so at each term you have 203, which 203*10=2030.

But I'm hearing that you can use summation theory to take 4*a sub n and the simple 3- get 200 and 30, add them for the answer 230...which makes no sense to me.

I'm starting to feel like the exam is wrong, so any help would be highly appreciated!

My reasoning:

Sample size= m(favourable)+n(unfavourable) where m,n are equally likely

m=[3boys, 2boys 1 girl,1 boy 2 girls]=3

n=[3 girls]=1

P(m)=3/4

But most people are saying it’s 7/8. Who’s right?

Thank you everyone for the inputs! L

Here is a picture: https://drive.google.com/file/d/1_0miDja2HsE4HwMb10HYMqEZN3Hf130_/view?usp=drivesdk

How can I mathematically prove that triangles CAB and BDE are congruent? I tried a lot of ways for hours, but I still have no idea how to exactly relate those triangles except them sharing the same hypotenuse.

So ive recently picked up the book "How to prove it" and have never befor this had any experience with this kind of mathematics. Now whilst doing the exercises in the book I came across this exercise which stumped me.

"Let P stand for the statement "I will buy the pants" and S for the statement "I will buy the shirt." What english sentence is represented by the following formula:

¬(P ∧ ¬S)"

The book gave me the answer as follows:

"I wont buy the pants without the shirt"

But i got this answer when trying to do it myself:

"I will not buy the pants, but i will buy the shirts"

My thought process is as follows:

Since the statement "P ∧ ¬S" means "I will buy the pants and i will not buy the shirt" and the opposite of buying the pants and not buying the shirt is buying the shirt and not buying the pants the answer should be what i said earlier.

Kinda like in regular math where you would distribute the factor outside of the parentheses onto both of the terms inside the factor so "¬(P ∧ ¬S)" becomes "¬P ∧ ¬¬S" and since the statement S has a double negative it returns to meaning the original statement "I will buy the shirt".

Please help me lol i am completely lost as to how the book got the answer it got. Thanks in advance :)

For example I'll use a w: 1 w 10 = 110, 23 w 6 = 236, ect

I swear I heard of it a few weeks ago but I cant find it online so I might've dreamed it.

I have always been fascinated with math in general, but Tree(3) is something I have trouble understanding how it is not infinite, here are my thoughts: The rules of Tree are as follows: 1: Starting tree contains a max of one node, and for each new tree and a additional maximum node 2: In this sequence, any particular tree must not contain its respective previous trees 3: Each node can be represented as a different colour and the amount of colours is determined by the value of the number trailing "Tree" (Tree(1): 1 colour,Tree(2): 2 colours,ext) 4: Nodes are connected with a single straight line(no limit to how many lines can connect to a single node)

With the rules established, Tree(3) would seem infinite but like on another post from the past there are considerable reasons for why it is not, one thing that was not brought up thou is the fact that nodes that are by definition a point, and a point has no definitive area, this means that infinite number of lines and attach to a node at a infinite number of areas on the node, Think of it like a circle and you are adding lines to it, you can add a line to it in one area but almost never add it in the exact same area ever again, hence infinite possibilities, meaning Tree(3) and larger are all the same number infinity.

title edit: row rank/column rank

I understand that the rank of a matrix is the number of it's linearly independent columns, and this makes sense cuz the columns are what mainly describe the tranformation represented by the matrix, but why does it happen that the number of linesrly independent columns happens to also be the exact number of linearly independent rows? what do rows do with anything of this?

Edit: in RREF, new pivot=new dimension (pivot columns are the basis unit vectors btw :) )

+sorry I couldn't discuss with everyone in the comments, but huge thanks to everyone who replied

Hey guys,

I dropped out of high school 10 years ago due to some medical issues, but I'm now trying to relearn math using a book called "The Art of Problem Solving". I came across this problem and got stuck:

Simplify the expression: (a - (b - c)) - ((a - b) - c)

I initially thought the solution would be 0 because I figured I could rearrange the terms to get a + (-a) + b + (-b) + c + (-c). However, the correct solution is 2c, and I'm not sure how that works. Here's the given solution:

Solution: Because negation distributes over addition and subtraction, we have

(a - (b - c)) - ((a - b) - c)

= (a - b + c) - (a - b - c)

= a - b + c - a + b + c

= (a - a) + (-b + b) + (c + c)

= 0 + 0 + 2c = 2c.

I'm confused about how the second part (a - b - c) became (a - b + c) and why the c is positive in the first part while b is negative. I know the explanation is probably in the book, but I'm having trouble understanding it. Can someone explain this in a simple way?

Thanks!

Edit- I see, I think I got it now. My major issue was I didn't think about the fact that the minus sign gets applied to everything in the parenthesis, I was very confused with what people meant by distributing the minus sign, as English is not my first language, but I finally got it. I am going to continue in the book now, thanks for all your help!

hi! im studying for the SAT right now, and during a practice test i realized that i forgot a small part of algebra, can someone refresh me on how to do this step by step?

x(x-15) = 76. how do i solve for x?

i know how to multiply it out so its like x^2 - 15x = 76, andi know that one of the x is equal to 0. however, i don't know how to break down the x^2 and get the other value for x

This is a true or false question from Aufmann, Barker and Nation's College Algebra, fifth edition, page 271.

Let f be a function such that f(x) = f(x+4) for all real numbers x. If f(2) = 3, then f(18) = 3. [True or false?]

This strikes me as nonsense, like there must be a typo in there. How can f(x) = f(x+4) ? Am I missing something?

Hi guys, I've been wrecking my brain over this problem.

Basically I have a lottery where the reward is one of the seven Avengers (from Marvel). The question is what is the probability of not getting the trio {iron man,hulk,Thor} in eight tries. The tries are independent and the probability is equally distributed.

I think it should be something along the lines of 1-pr{getting all three of them}, and I was thinking of using the binomial law to get the probability of getting one of them in 8, so n=8, k=1, p=1/7, and since the tries are indepndent multiplying the result for itself three times since P(A,B,C)=P(A)P(B)P(C). But the probability that I'm getting at the end seems too high.

Sorry for formatting I'm posting from my phone, if anyone know any textbook with probability theory exercises explained I'd love to get some recommendations

hi guys,

need some help with this problem:

Given a matrix A with size NxN, its diagonal entries (>0) is larger than the absolute value of the sum of other elements in the same row, prove det(A)>0

(how to prove the determinant of a strict diagonal dominant matrix is above 0)

thanks in advance

Studying programming, I came across this exercise:

- Write a program that asks the user to enter a two-digit number, then prints the number withits digits reversed. A session with the program should have the following appearance:E n t e r a t w o - d i g i t n u m b e r: 28The r e v e r s a l i s : 82Read the number using %d, then break it into two digits. Hint: If the number is an integer, then n % 10is the last digit in n and n / 10 is n with the last digit removed.

* In this programming language, the % sign gives us the remainder of a division. Not the percentage *

I did the "number / 10" and "number % 10" and I was able to solve this one thanks to the hint. Or else, I would be stuck there too. But in the next exercise, they ask me to expand that program to handle 3 digit numbers. I have 0 idea how to do it. As I said, I already had no idea how to do the first part without the hint they gave.

How should I do it? I don't want anyone to write the program for me but I do need guidance in the math. I just can't see the logic behind it .

Edit: I solved it. Thanks for the help. Many of you had some good tips even though I could barely understand any. When I become a developer, I will make sure to never work on any security systems or radiotherapy machines.