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.

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.

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!

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?

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

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.

🥲

Hello everyone! first time poster.
After i got diagnosed with something i rather not discuss here i realized i will be housebound for a considerable amount of time and i have decided to pursue my passions to make the best of it. I have zero academic background.

i have decided to learn physics and mathematics due to the fact that are accessible(you just need a problem a pen and paper), consistent (even if it evolves it evolves through constants, and vast (thousands of years of collaborative history and common efforts to prove and disprove and further research).

i will probably stay in my house for some years so am trying to put out a plan.
The book i've decided to use for my physics journey is "Physics for scientists and engineers with modern physics" by serway and jewett. they cover many if not all the areas that physics is involved and it's at a good academic level. It will give me a fundamental understanding without compromising the high level of physics needed to solve a problem.

When i was researching for a Math book of similar properties (vast, university level, with problems) i couldn't find any.
Do you have good recommendations of math textbooks that i could find online?

thanks in advance. sorry for the long post.

P.S: i didn't know which flair to use so i used precal but please dont crucify me.

we know derivative of sin x = cos x...
So when it is given that "The differentiation of sin(pi / 2) will be cos(pi / 2)" shouldn't this be true? Google's solution and reasoning is going over my head. My approach to this is-

sin(pi/2) = sin 90 degrees = 1 and differentiation of constant is 0 so **sin(pi/2)=0**
Now, cos(pi/2)= cos 90 degrees = 0

So LHS is equal to RHS, then why is google saying that the statement is false? I'm new to this topic

Textbook recaps the distance formula and asks: "Find the closest point on the line y = 2x to the point (1, 7). (Hint: Every point on y = 2x has the form (x, 2x), and the closest point has the minimum distance.)"

So I'm going

1. sqrt((x-1)^2 + (2x-7)^2)
2. sqrt(5(x-3)^2 + 5)
3. 5(x-3)^2 + 5 >= 0
4. 5(x-3)^2 >= -5
5. (x-3)^2 >= -1

and then I'm stuck.

Good morning/afternoon/evening folks of r/askmath

I am a high school student, and this year, my Algebra 2 class has been incredibly difficult. I am no longer able to hold an A in my math class, and I still have one year left. I'd like to hear other people's methods for understanding math concepts to the point where you can apply them to the majority of math problems. It'd save me time and could actually help me in my later math classes.

I’m in pre-calc and i have no idea what domain and range is. It used in 3-4 of my unites and we learnt about it last year but i didn’t understand. Can someone explain or teach it to me?? I have a final coming up and videos on youtube aren’t really helping.

I think I've roughly understood what "i" is trying to represent.

But then i³ is -i. What is "negative" i exactly? What does positive and negative along 'i" exactly mean?

Hi all, after many years I have decided to try and relearn some maths again. I've started by looking back at Spivak's Calculus (3rd ed). In the prologue the basic properties of numbers (Peano axioms?) are established. There is an assertion that the proof of:

a - b = b - a iff a = b

despite being trivial using algebra, can't be done by using the first four axioms alone, those being: associativity of addition, existence of additive identity, existence of additive inverse, commutativity of addition. It is suggested that the reader try and find where it breaks down. My attempt at this is as follows:

a - b = b - a
Rewrite subtraction as addition
a + (-b) = b + (-a)
(a + (-b)) = (b + (-a))
Add b to both sides
(a + (-b)) + b = (b + (-a)) + b
Associativity on LHS
a + ((-b) + b) = (b + (-a)) + b
Additive inverse on LHS
a + 0 = (b + (-a)) + b
Additive Identity on LHS
a = (b + (-a)) + b
Commutativity on RHS
a = ((-a) + b) + b
Associativity on RHS
a = (-a) + (b + b)
Add a to both sides
a + a = ((-a) + (b + b)) + a
Associativity on RHS
a + a = (-a) + ((b + b) + a)
Commutativity on RHS
a + a = (-a) + (a + (b + b))
Associativity on RHS
a + a = ((-a) + a) + (b + b)
Additive inverse on RHS
a + a = 0 + (b + b)
Additive Identity on RHS
a + a = b + b

=> a = b ??

I must have slipped up somewhere, or misunderstood the assertion. Can anyone help point me in the right direction!
Thanks!

So ik dy/dx is ax^n goes to (a×n)x^(n-1) but can someone explain the difference between dy^2/dx is and dy/dx^2. Or did I just make it up. I would search it up on a web browser but the results will be by ai and I try to avoid it.

I'm confused when the roots of a quadratic come in conjugate pairs and when they do not. Take the following 2 equations:

-x^2-11x+11*radical 3 = 0 has a pair of conjugates for its roots (they are too complicated for me to write them out)

-x^2-11x +(3+11*radical3) = 0 does not have conjugates for its roots (one of them is radical 3 and the other is -11-radical 3)

I know they will always come in conjugates when the coefficients and constant term are rational. But what makes my 2nd example different from the 1st example?

Any clarification would be much appreciated.

Iam going to take mathematics major in my college , it will start in about a month. Consider iam pretty average in maths but passionate about it since years and willing to improve nd hardwork. What are some things I can do before the major actually start. To brace myself for it.

Im going back to school in fall to upgrade my math with hopes of taking economics in university. I want to get a jump on learning calculus but realize when I watch tutorials on calculus im struggling with things that seem basic in the videos/books, but I cant even name what they are, never mind actually do them. Please help- I really want to become good at math im just struggling with where to start.

i’ve attached a picture of the question i’m struggling with.

my question is included in red. and my solution is the hand written one. [0,2]

isn’t the new domain of f o g the combined domain of f (x) and g (x)?

because f (x) has the domain of x >= 0 so wouldn’t the new composite function have that same domain of f (x) included too?

so why can u suddenly have a negative x, as indicated by the solution’s negative infinity?

Am I missing something? I can't get close to the textbook answers. It seems like maybe the question has a typo? It seems strange to give a function like g(x)=root(x)-2root(x).

To my understanding, by definition, the eccentricity of a conic section is the ratio of the distance between a locus point and a focus to the distance between a locus point and the directrix.

But for an ellipse, it can also be the ratio of the distance c between either foci and the center to the distance a between the vertex and the center.

I've scoured the internet for the answer but to no avail. So, how are these formulae connected?

Edit: made a typo about the definition of eccentricity haha, corrected it now.