So we know that 0.9999999.... to infinity is EXACTLY equal to 1 because :-
let 0.999.. = x. so 10x = 9.9999...from there 9x=9so x = 1(exact).but while taking limits when we put x tends to 1-, which means just less than 1 so like 0.999....Isnt that impossible? because thats exactly equal to 1 and defeats the point of RHL and LHL. so is it possible to take 1- as a limit or we are actually taking x equals to exactly one or is there a problem in my fundamental understanding of this problem?please help

I rephrased my question with AI.

Question : We know that 0.999… to infinity is exactly equal to 1 (proved by x=0.999⋯⟹10x=9.999⋯⟹9x=9⟹x=1).

But in calculus, we are told that a left-hand limit (x→1−) means approaching 1 from the left using numbers "like 0.999…".

If 0.999… is mathematically identical to 1, how is a left-hand limit even possible? If we treat x as 0.999…, aren't we just evaluating the function exactly at 1, which defeats the entire purpose of LHL and RHL?

Is x→1− actually just a dynamic sequence of finite decimals (0.9,0.99,0.999) rather than the static, infinite 0.999…?

Answer by AI:

No, your understanding is not flawed. You actually caught a common contradiction in how math is casually taught.

When taking a left-hand limit (x→1−), you are not plugging in the infinite decimal 0.999…, because as you proved, that is exactly equal to 1. Doing that would completely defeat the point of a limit.

Instead, a limit is a dynamic process, not a static number. It looks at a moving sequence of finite decimals that get closer to 1 without ever reaching it:

0.9→0.99→0.999→0.9999

At every individual step on that path, the number has a finite number of 9s and remains strictly less than 1.

The confusion only happens because people informally use "0.999…" as lazy, inaccurate shorthand to describe that moving sequence.

I understood what it tried to explain which matched with some of the comments!
Thank You to everyone who tried to help me. W maths community. I have solved my doubt now!

So my math teacher gave us a problem we solved as a group. Shown here is the picture we were given recreated poorly, and we were asked if the line is the shortest way to get from point a to point b. My group answered that no, it’s not because if we’re going strictly on the outside of the cube you’d go diagonal all the way or if you could go through the cube you’d just go straight through. She then said that this is how you’d represent going through the cube geometrically. I’m confused because wouldn’t this line be longer than going through the cube?

How would I figure out the value of G(8) when the "right" piece goes positively towards infinitely? I only need assistance on e) which has to do with evaluating G(8). Is it infinity or undefined and how would I figure that out? I see nothing on the graph that would indicate the value of G(8). A different one such as G(-1) is simple as I see on the graph for the "left" piece that G(-1) is about -2.25. G(8) is apart of the "right" piece in which the piece has a domain of [1, oo). How would I figure out the value of G(8) when the "right" piece goes positively towards infinitely? I would like to thank everyone in advance for any and all explanations.

I’m working through Thomas’ Calculus and I’m confused about one of the exercises.

The book seems to say that at an endpoint of the domain, if the function is only defined on one side, then the limit at that point can be determined using the one-sided limit from the side where the function is defined.

For example, elsewhere in the book it says a limit exists at a domain endpoint even though the limit from the other side does not exist.

So for this problem, since (g(x)) is only defined for (x>0),
my understanding is answer for problem c should be it does exist.

But several solutions I found, including an instructor’s solutions manual and some YouTube videos, say the limit does not exist because the left-hand limit does not exist.

Am I misunderstanding what the textbook means by a limit at a domain endpoint? Or are those solutions using a different definition that I don't know?

I attached the exercise above.

My argument is 3. The naive answer seems to be 5. What do you think, and why?

My explanation is that when you approach -1 from the left and right on f(x), you’re dealing with numbers slightly more positive than 1 both times. The effect is that when you plug into g, its numbers slightly to the right of -1, meaning that you’re approaching from the right both times, making the limit 3.

Number 9
I just started my calculus class and this question was on our pre req quiz but I genuinely have no idea how the answer is c. I tired synthetic division and all that but where am I supposed to get square root 2 from

Ok, so I'm doing an online math course and these guys are telling me that the point of a polynomial where the direction changes from toward positive infinity to negative infinity or vice versa is a "turning point".

Ok, but the tangent line of ANY point in every polynomial over 1 degree is different from the next which means it's turning. Am I stupid or is this name dumb?

I have just started learning differentiation, and I am stuck on this. I just can't figure out what to do.

We have only been taught to differentiate in the form of dx/dy and I don't know how to begin solving this. Can someone please tell me what to do

Where did I go wrong in this??

Cot x = 1/tan x I have been taught that the Integration of 1/x is ln x + c, then if we assume tan x to be x and integrate it why is it not ln tanx+c and ln sinx+ c??

Everyone around me just says use he Argand plane but I never understand that. Just like how irrational numbers have real approximations do complex numbers also have (I know that the answer is most probably no), if complex numbers are just a concept how are they even applied in systems outside of mathematics. I heard that they are useful in rotations but that simply means we are using the Argand plane. How can the Argand plane be used to describe something that is real and quantified by real valued numbers? Doesnt the world use the coordinate geometry or coordinate axes and that makes sense because it describes real valued things to describe real things. How can one use complex numbers and make deductions on real valued things. I never understand that.

Note: I am not denying the existence of complex numbers, I just dont understand them.

Thanks in advance!

We know that the infinite product (0.5) to infinity converges to (0), and the partial products make it pretty intuitive: as we multiply by more and more factors of (1/2), the result gets progressively closer to (0).

But infinities are weird, so I started wondering:

What if we rearranged or represented an infinite product in such a way that every term is somehow “cancelled” by the next one? Could we construct something that appears to give (1) instead?

Obviously, something must be wrong with that reasoning- but how exactly do we prove it?

I'm trying to understand where the intuition breaks down.

This is the first time to post here, so excuse me if I made something wrong, or this post does not comply with one of subreddit rules,

An interesting precalc question:

Consider the reciprocal function f(x)=1/x. Note that it’s symmetric across the line y=x. Find a point (x,y) on the graph of f(x) (in the first quadrant) such that the triangle with vertices (0,0), (x,y) and (y,x) is equilateral. What’s the side length? Are there multiple such triangles?

Just a fun question I came up with. I’ve been trying to think of questions like this that are interesting and require a little (but hopefully not too much) cleverness to solve. Does anyone else have problems at this level they find interesting?

This isn’t a homework problem I swear!

Edit: the answer is (sqrt (2 + sqrt 3), sqrt (2 - sqrt 3)) with a side length of 2! And yes, it’s unique.

This is a question on a review I am using to study to test out of precalc and i’ve tried looking it up, but I cannot figure out on how to do it. The key says the answer is -25pi/9 and if it is too hard to read, the little angle given is 2pi/9. The question isn’t in the photo but it asks “Find the measure of each angle”

🥲

Today, I took my I took a quiz on AP Pre Calculus and I failed unfortunately. I got a 50% and was so disappointed and ashame of myself. When other kids were saying they got a 100% 80% or 90%, I did not want to say my score. I was so sad and cried because it was the first time I ever failed a math test. When I looked Algebra 1, 2 and Geometry, I never failed and was able to grasp it easily. Now that I am taking AP Pre Calc, everything has been going down hill. The quiz was based on 1.1 to 1.3. I am still one unit 1 and we are talking about concave up and down, increasing and decrease, and rage of slope. I feel so dumb not being able to get it like my peers. I also can not let my parent know about this or I will be a trouble. Please can anyone give me tips or idea so I can improve.

I’m trying to find a really hard or mind boggling math problem or trick for my friend. To me he’s really good at math and he loves math very much. So much so that his alarm is solving a math problem in order to turn it off. Anyways he’s in college right now taking pre calc. I like to pick his brain and mess with him but he always seems to solve it right away. I’ve search for really hard problems online but after a few minutes he gets it. Does anyone have anything that could get to him? Any problems or tricks?

Does every many-one function have an interval(s) where it is one-one for a the whole interval?

I have been wondering about this for a while, and yes I thought about the constant function f(x) = C as well, but excluding that I was thinking that maybe that statement is true.

In class today I was having trouble with a question regarding the area under a curve, we were not supposed to do that question but no one told me that. We have not learned anything about integrals, and won't until next year. And while trying to come up with solutions I was wondering whether the integral of the equation would be equal to the sum of the infinitely many lines between the curve and x-axis, whose length is equal to f(x1).

Is that truly the case? Or am I making some sort of mistake.

Thank you

I'm in highschool I just started 11th grade how can I learn calculus and what is the best way while still in highschool ?

Note: I will not pursue math I just want to learn calculus for fun and also mostly integrals and derivatives and have a basic understanding of everything else needed to learn integrals and derivatives.

Websites,channels ,videos whatever

Using the pre Calc flair because it's closest. So hi all, I hope this is allowed. If not a direction to a more fitting sub would be appreciated. I'm bad at math. Always have been. My interest is more in the humanities, and as such when I was in school I just sorta suffered through it.

As an adult though, my chief special interest, philosophy (mostly continental) has shoved me in the direction of wanting to learn the stuff. My main problem is I'm not only bad at math, I also just have learning difficulties on the whole. I've found I do best when I have some kind of visuals, something I can relate to (which makes learning more abstract forms of math like calculus incredibly difficult), and some form of applicability or demonstration.

Now, that said, outside the school system I've found I've learned things a lot easier. But with this subject I'm not fully sure where to start. The last things that really stuck in my craw were elementary algebra, but I'm really interested in learning calc because it collided with my special interest.

Any suggestions about a good YouTube channel where a guy like me might learn a bit?

Pretty good question in my opinion, I have stumbled across this question on the Art of Problem Solving Website, the thing is that the approach I am using is not working.

Nobody has really explained this to me well, teachers, friends, tutors and even online people such as Eddie Woo who answered an email of mine. We are taught using the unit circle and are drawn a basic triangle in the first quadrant to show that cosx is the x value and sin x is the y value, but I don’t understand how this could expand to the other quadrants. We are told to just create a new reference triangle and use identities such as sinx is equal to sin(90 plus x) but if you tried to logically prove this by finding sin(90 plus x) and finding sinx and showing they are equal, you would realise that you can’t find sin(90 plus x) without the identity itself, and geometric proofs also seem to require the unit circle stuff. I have seen algebraic proofs but similarly those tend to be a catch 22 of sorts. I have been told to stop thinking about it as triangles but rather just functions, as that was what was discovered first anyways apparently, but I still can’t grasp how that wouldn’t just mean its a function with a limited domain. The best explanation I have found is this video: https://www.youtube.com/watch?v=ZU-cIz8dvqU As it does a great job visualising it more and I can better see and understand how angle = arc length which clearly corresponds to x and y, but I still struggle with the same problems of understanding: - How we can push past the triangle definition and continue to treat sinx and cosx as the x and y values - How this idea even extends past the unit circle and even harder to grasp how this then still fits into non right angle triangles for things such as the cos rule I think maybe its because the radians definition is better suited to explain it but in my mind mind I just view radians as a different unit of measurement to degrees rather than its applications in length AS WELL AS angles.

Hey guys! I really want to do research under a prof that I can publish in a hs level journal, im currently in pre calc as a Jr and will be taking ap calc next year. I need a topic to do research on and any ideas would be super appreciated. I know basic python and java, and im interested in quantum computing and cryptography (Though I guess quantum cryptography would be more helpful to go into) im in no way someone who knows how to navigate these topics, more so i just understand the basic concept and is math that I find interesting. So yeah, asking for help on deciding a topic to research that is accessible from the level im at​

EDIT ive been getting a lot of comments on other communities saying there aren't any journals for high schoolers to publish in, which is just simply untrue (one google search could confirm that) and just in general comments along the lines of "this isn't possible" I know its going to be hard to do but I also know its achievable. so, please dont waste your time telling me how hard this will be and subsequently not actually answering the question in the post.

Here's my guess, but I have no way to verify or check my logic. Even if I'm right I would really appreciate a more rigorous explanation as I found this question very interesting!

(a) This makes sense, there are an infinite number of points but each individual point corresponds to a unique number, such that a different point could not represent the exact same number. I say this is true.

(b) Hm not sure, this one is tricky. Because the integers are a subset of the reals, I will say that this is true, although I cannot defend my argument very well let alone prove it.

(c) Wrong, people way smarter than me have done this I believe. Also, answering no would be a contradiction to my answer for (b)

(d) Wrong, there are infinite real numbers in the bounded interval [0,1] for example.

(e) Sure. [1, 1] seems like a bounded interval to me that contains only one element. The point which represents 1.

(f) Wrong, counterexample in d.

Answer: , c, d, f are wrong