Art of Problem Solving
AoPS Online AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy AoPS Academy
Small live classes for advanced math
and language arts learners in grades 1-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Summer is a great time to explore cool problems to keep your skills sharp!  Schedule a class today!
No tags match your search
M
G
Topic
First Poster
Last Poster
H
k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!

MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 1, 2025
0 replies
J
H
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
J
H
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
J
H
IMC 1994 D2 P3
j___d   4
N 30 minutes ago by krigger
Let $f$ be a real-valued function with $n+1$ derivatives at each point of $\mathbb R$. Show that for each pair of real numbers $a$, $b$, $a<b$, such that
$$\ln\left( \frac{f(b)+f'(b)+\cdots + f^{(n)} (b)}{f(a)+f'(a)+\cdots + f^{(n)}(a)}\right)=b-a$$there is a number $c$ in the open interval $(a,b)$ for which
$$f^{(n+1)}(c)=f(c)$$
4 replies
j___d
Mar 6, 2017
krigger
30 minutes ago
J
H
number theory problem
danilorj   1
N 2 hours ago by solidgreen
Let $t$ be an integer, show that there are infinite perfect squares of the form $3t^2+4t+5$
1 reply
danilorj
Today at 1:37 PM
solidgreen
2 hours ago
J
H
Combo resources
Fly_into_the_sky   1
N 4 hours ago by Fly_into_the_sky
Ok so i never did combinatorics in my life :oops: and i am willing to be able to do P1/P4 combos (or even more)
So yeah how can i start from scratch?
Remark:i don't want compuational combo resources :noo:
1 reply
Fly_into_the_sky
4 hours ago
Fly_into_the_sky
4 hours ago
J
H
2023 Putnam B2
giginori   13
N 4 hours ago by pie854
For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 \cdot n$. What is the minimum value of $k(n)$?
13 replies
giginori
Dec 3, 2023
pie854
4 hours ago
J
H
Very odd geo
Royal_mhyasd   2
N 5 hours ago by Royal_mhyasd
Source: own (i think)
nevermind
2 replies
Royal_mhyasd
Yesterday at 6:10 PM
Royal_mhyasd
5 hours ago
J
H
Polynomial Application Sequences and GCDs
pieater314159   46
N 5 hours ago by cursed_tangent1434
Source: ELMO 2019 Problem 1, 2019 ELMO Shortlist N1
Let $P(x)$ be a polynomial with integer coefficients such that $P(0)=1$, and let $c > 1$ be an integer. Define $x_0=0$ and $x_{i+1} = P(x_i)$ for all integers $i \ge 0$. Show that there are infinitely many positive integers $n$ such that $\gcd (x_n, n+c)=1$.

Proposed by Milan Haiman and Carl Schildkraut
46 replies
pieater314159
Jun 19, 2019
cursed_tangent1434
5 hours ago
J
H
c^a + a = 2^b
Havu   10
N 5 hours ago by Havu
Find $a, b, c\in\mathbb{Z}^+$ such that $a,b,c$ coprime, $a + b = 2c$ and $c^a + a = 2^b$.
10 replies
Havu
May 10, 2025
Havu
5 hours ago
J
H
Own made functional equation
JARP091   0
5 hours ago
Source: Own (Maybe?)
\[ \text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\ f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right) \]
0 replies
JARP091
5 hours ago
0 replies
J
H
Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   16
N 6 hours ago by JARP091
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
16 replies
OgnjenTesic
May 22, 2025
JARP091
6 hours ago
J
H
equal segments on radiuses
danepale   8
N 6 hours ago by zuat.e
Source: Croatia TST 2016
Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.
8 replies
danepale
Apr 25, 2016
zuat.e
6 hours ago
J
H
Inequality
SunnyEvan   8
N 6 hours ago by arqady
Let $a$, $b$, $c$ be non-negative real numbers, no two of which are zero. Prove that :
$$ \sum \frac{3ab-2bc+3ca}{3b^2+bc+3c^2} \geq \frac{12}{7}$$
8 replies
SunnyEvan
Apr 1, 2025
arqady
6 hours ago
J
H
Inequality conjecture
RainbowNeos   2
N 6 hours ago by RainbowNeos
Show (or deny) that there exists an absolute constant $C>0$ that, for all $n$ and $n$ positive real numbers $x_i ,1\leq i \leq n$, there is
\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]
2 replies
RainbowNeos
May 29, 2025
RainbowNeos
6 hours ago
J
H
2- player game on a strip of n squares with two game pieces
parmenides51   2
N 6 hours ago by Gggvds1
Source: 2023 Austrian Mathematical Olympiad, Junior Regional Competition , Problem 3
Alice and Bob play a game on a strip of $n \ge 3$ squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of $ n = 7$ squares.
IMAGE
The players alternate. In each move, they advance their own game piece by one or two squares in the direction of the opponent’s piece. The piece has to land on an empty square without jumping over the opponent’s piece. Alice makes the first move with her own piece. If a player cannot move, they lose.

For which $n$ can Bob ensure a win no matter how Alice plays?
For which $n$ can Alice ensure a win no matter how Bob plays?

(Karl Czakler)
2 replies
parmenides51
Mar 26, 2024
Gggvds1
6 hours ago
J
H
D1040 : A general and strange result
Dattier   0
Today at 12:46 PM
Source: les dattes à Dattier
Let $f \in C([0,1];[0,1])$ bijective, $f(0)=0$ and $(a_k) \in [0,1]^\mathbb N$ with $ \sum \limits_{k=0}^{+\infty} a_k$ converge.

Is it true that $\sum \limits_{k=0}^{+\infty} \sqrt{f(a_k)\times f^{-1}(a_k)}$ converge?
0 replies
Dattier
Today at 12:46 PM
0 replies
J
H
J
H
A very simple question about calculus for middle school students
Craftybutterfly   19
N Apr 15, 2025 by Craftybutterfly
Source: own
$\lim_{x \to 8} \frac{2x^2+13x+6}{x^2+14x+48}=$ ? (there is an easy workaround)
(I know this is very easy- a little child can solve this in 1 second kinda problem so don't argue or mock me please)
19 replies
Craftybutterfly
Apr 9, 2025
Craftybutterfly
Apr 15, 2025
J
A very simple question about calculus for middle school students
G H J
Reveal topic tags
calculus
L
G H BBookmark kLocked kLocked NReply
Source: own
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Craftybutterfly
589 posts
Craftybutterfly
#1 h  Apr 9, 2025, 7:41 AM
Y by
$\lim_{x \to 8} \frac{2x^2+13x+6}{x^2+14x+48}=$ ? (there is an easy workaround)
(I know this is very easy- a little child can solve this in 1 second kinda problem so don't argue or mock me please)
This post has been edited 2 times. Last edited by Craftybutterfly, Apr 10, 2025, 9:26 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Craftybutterfly
589 posts
Craftybutterfly
#2 h  Apr 9, 2025, 7:44 AM
Y by
sol
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Yiyj1
1268 posts
Yiyj1
#3 h  Apr 9, 2025, 7:49 AM
Y by
Click to reveal hidden text
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GentlePanda24
746 posts
GentlePanda24
#4 h  Apr 9, 2025, 2:33 PM
Y by
solution
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
KAME06
161 posts
KAME06
#5 h  Apr 9, 2025, 4:10 PM
Y by
We can plug because the function is continuous in $8$.
Pretty well known that polynomials are continuous, and $2x^2+13x+6$ and $x^2+14x+48$ are continuous on $8$ and different from $0$ when you evaluate them on $8$.
That implies that $\frac{2x^2+13x+6}{x^2+14x+48}$ is continuous on $8$, so $\lim_{x \to 8} \frac{2x^2+13x+6}{x^2+14x+48}=\frac{2(8)^2+13(8)+6}{(8)^2+14(8)+48}$ (factorize if u want to do less work).
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
HacheB2031
408 posts
HacheB2031
#6 h  Apr 10, 2025, 5:22 AM
Y by
Craftybutterfly wrote:
Find $\lim_{x \to -6} \frac{2x^2+13x+6}{x^2+14x+48}.$

FTFY
This post has been edited 1 time. Last edited by HacheB2031, Apr 10, 2025, 2:37 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Craftybutterfly
589 posts
Craftybutterfly
#7 h  Apr 10, 2025, 5:24 AM
Y by
HacheB2031 wrote:
Craftybutterfly wrote:
Find $\lim_{x \to -8} \frac{2x^2+13x+6}{x^2+14x+48}.$

FTFY
?
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Craftybutterfly
589 posts
Craftybutterfly
#9 h  Apr 10, 2025, 6:02 AM
Y by
@sp0rtman00000Click to reveal hidden text
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
HacheB2031
408 posts
HacheB2031
#10 h  Apr 10, 2025, 2:37 PM • 1 Y
Y by LawofCosine
Craftybutterfly wrote:
HacheB2031 wrote:
Craftybutterfly wrote:
Find $\lim_{x \to -6} \frac{2x^2+13x+6}{x^2+14x+48}.$

FTFY
?
I'm sorry, I realized it should probably be as $x\to-6.$ Let me explain why.
When you make limits, you often want them to tend to a value where a function is not defined. Take the limit \[\lim_{x\to0}\frac{\sin x}x.\]Clearly, direct evaluation gives \[\frac{\sin0}0=\frac00,\]an indeterminate form, but (using methods such as the squeeze theorem) you can show that \[\lim_{x\to0}\frac{\sin x}x=1.\]Despite the function not being defined there, it has a well-defined limit. In my [edited] post, notice that as $x\to-6,$ you also have $x^2+14x+48\to0$ and $2x^2+13x+6\to0,$ making this limit not evaluatable with direct evaluation. (The limit may exist, but in the old post where $x\to-8,$ it blows up.) If you have the limit of a rational function, a function where the numerator and denominator are both polynomials, you can cancel common factors to redefine the function where it wasn't defined. Also, note that you can't always use direct evaluation. Let \[f(x)=\begin{cases}x^2&x\ne0\\1&x=0\end{cases}.\]Try to evaluate the limit \[\lim_{x\to0}f(x).\]If we directly evaluate it, we get that \[\lim_{x\to0}f(x)=f(0)=1.\]However, $f(x)$ has a problem: it isn't continuous, or, if you drew its graph, you would need to pick up your pencil from the paper. The actual limit turns out to be \[\lim_{x\to0}f(x)=\lim_{x\to0}x^2=0.\]Because $x^2$ is a polynomial, it is continuous; all polynomials are continuous. As well as all rational functions, over their domain. Anywhere a rational function is defined, it is continuous, except at points where its denominator is $0.$ sidenote
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Craftybutterfly
589 posts
Craftybutterfly
#11 h  Apr 10, 2025, 6:19 PM
Y by
It is $\lim_{x\to8}$ not $\lim_{x\to-6}$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
yaxuan
3425 posts
yaxuan
#12 h  Apr 10, 2025, 8:51 PM
Y by
op wrote:
a kindergartener can solve this in 1 second
Bruh
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Craftybutterfly
589 posts
Craftybutterfly
#13 h  Apr 10, 2025, 9:27 PM
Y by
Craftybutterfly wrote:
sol
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Oshawoot
147 posts
Oshawoot
#14 h  Apr 14, 2025, 2:27 AM
Y by
whats lim again？。。。
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
LawofCosine
862 posts
LawofCosine
#15 h  Apr 14, 2025, 2:40 AM
Y by
a limit (concept in calculus)
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GentlePanda24
746 posts
GentlePanda24
#16 h  Apr 14, 2025, 1:11 PM
Y by
Craftybutterfly wrote:
a kindergartener can solve this in 1 second

I think that is a bit exagerated
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
GentlePanda24
746 posts
GentlePanda24
#17 h  Apr 14, 2025, 1:12 PM
Y by
Craftybutterfly wrote:
a kindergartener can solve this in 1 second

I think that is a bit exagerated
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
leahlyoung106
63 posts
leahlyoung106
#18 h  Apr 14, 2025, 2:12 PM
Y by
Click to reveal hidden text
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Craftybutterfly
589 posts
Craftybutterfly
#19 h  Apr 15, 2025, 5:24 AM
Y by
Question: what is an example of a with no limit or a limit of 0
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
aidan0626
1965 posts
aidan0626
#20 h  Apr 15, 2025, 5:29 AM • 1 Y
Y by LawofCosine
Craftybutterfly wrote:
Question: what is an example of a with no limit or a limit of 0

x-> -8 has no limit
x-> -1/2 gives a limit of 0
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Craftybutterfly
589 posts
Craftybutterfly
#21 h  Apr 15, 2025, 5:35 AM
Y by
Thx @bove
Z K Y
N Quick Reply
G
H
=
a