,
,
,
,
or even expressions of those terms combined as if that would make them defined. I have made this post to answer these questions once and for all, and I politely ask everyone to link this post to threads that are talking about this issue.
,
isn't even rigorously defined in this expression. What exactly do we mean by 
.
,
does not exist in the first place, although you will see why in a calculus course. So the point is that 
?
indeterminate? In analysis (the theory behind calculus and beyond), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits. However, if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. For example, we could say that 
,
or ![\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]](http://latex.artofproblemsolving.com/2/6/0/260f97a1cf1ea8722ff97a29b83bac7bfe83e577.png)
etc etc. So what makes something undefined? In the broader scope, something being undefined refers to an expression which is not assigned an interpretation or a value. A function is said to be undefined for points outside its domain. For example, the function 
given by the mapping 
is undefined for 
.
is undefined because dividing by ![\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]](http://latex.artofproblemsolving.com/9/9/3/993c78b62ec2c84b5d77daa8e9a8cd6f70fcf904.png)
So instead of treating infinity as an idea, we define infinity (positively and negatively, mind you) as actual numbers in the reals. The advantage of doing this is for two reasons. The first is because we can turn this thing into a totally ordered set. Specifically, we can let 
for each 
which means that via this order topology each subset has an infimum and supremum and 
is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In ![\begin{align*}
a + \infty = \infty + a & = \infty, & a & \neq -\infty \\
a - \infty = -\infty + a & = -\infty, & a & \neq \infty \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\
\frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\
\frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\
\frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0).
\end{align*}](http://latex.artofproblemsolving.com/6/d/8/6d8fc1eca5c791d430a7168cd3b37a02104fa318.png)
So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,![\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]](http://latex.artofproblemsolving.com/a/0/b/a0b67076efcc014c78b2023be1151c84b57c1503.png)
So while we define certain things, we also left others undefined/indeterminate in the process! However, in the context of measure theory it is common to define 
as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by ![\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]](http://latex.artofproblemsolving.com/8/4/e/84ed3d6ab237df970333b87beaef5311caa46185.png)
which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, 
means complex infinity, since we are in the complex plane now. Here's the catch: division by ![\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]](http://latex.artofproblemsolving.com/1/6/0/160cb939707708ff4e0930724df2928af11d7fd1.png)
where 
and 
Furthermore, we actually have some nice properties with multiplication that we didn't have before. In 
it holds that![\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]](http://latex.artofproblemsolving.com/8/e/0/8e035b4841d20c5ad671c46355cc667ca01e533b.png)
but 
and 
are left as undefined (unless there is an explicit need to change that somehow). One could define the projectively extended reals as we did with ![\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]](http://latex.artofproblemsolving.com/c/0/4/c044ff37849acad3b8b280b9753539225a1276ca.png)
They behave in a similar way to the Riemann Sphere, with division by 
be positive real numbers such that 
Find the minimum value of 
Prove that
and 
connect the points 
and 
These arcs lie in the same half-plane defined by line 
in such a way that arc 
lies between the arcs 
and 
Point 
lies on the segment 
Let 
,
be three rays starting at 
lying in the same half-plane, 
being between 
and 
For 
denote by 
the point of intersection of 
and 
(see the Figure below). Denote by 
the curved quadrilateral, whose sides are the segments 
and arcs 
and 
We say that this quadrilateral is 
if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals 
are circumscribed, then the curved quadrilateral 
is circumscribed, too.
Prove that
red marbles, 
yellow marbles, 
green marbles, and 
blue marbles. If Amanda takes the marbles one by one without replacing them until the 
marble. Then the probability that the 
marble is red is?
and 
that satisfy
and 
intersect at the point 
.
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
such that 
.
,
lies on diagonal 
.
.
has 
,
,
.
.
indicates my favorites.
is defined over the positive integers as follows: 
,
for 
prime, and for all relatively prime positive integers 
.
is the smallest positive integer such that 
,
,
,
,
,
,
?
and 
be the greatest odd divisor of 
for even 
denote the product of every odd positive integer less than 
for positive integers 
and 
where 
is minimized, find the number of divisors of 
.
consists of a line and a point not on the line. The distance from the point to the line can be expressed as 
,
.
.
are positive real numbers such that 
,
,
,
be a convex pentagon satisfying 
,
,
.
be the intersection of lines 
and 
.
has a perimeter of 
and an area of 
,
be an acute scalene triangle with longest side 
be the circumcenter of 
are chosen on 
and 
.
,
,
,
.
such that 
is divisible by 
by 
,
such that in every row and column, the number of 
or 
is valid, while a row with squares labeled 
or 
is not valid.) Find the remainder when 
.
,
be real numbers satisfying the system of equations
What is the value of 
?
at 
and 
are 
be the set of all 
,
Y by internationalnick123456
Is it possible to color all lattice points in plane into 3 colors such that
1. every line passing through lattice points and parallel to x axis has these three colors infinitely many(that is, every color appears infinitely many times in those lines).
2. every line passing through lattice points and not parallel to x axis cannot have three different color lattice points on it.
I think the answer is yes, but I couldn't find an example...
1. every line passing through lattice points and parallel to x axis has these three colors infinitely many(that is, every color appears infinitely many times in those lines).
2. every line passing through lattice points and not parallel to x axis cannot have three different color lattice points on it.
I think the answer is yes, but I couldn't find an example...
This post has been edited 5 times. Last edited by CBMaster, Apr 11, 2025, 6:58 PM
Reason: .
Reason: .
