,
,
,
,
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 
of 
-
and, for each 
,
strings 
for which 
and 
differ in exactly one character.
be three positive real numbers with 
Prove that![$$ab+bc+ca+a+b+c+3\ge \left[\frac{\sqrt{\left(a+1\right)\left(b+1\right)}}{2}+\frac{\sqrt{\left(b+1\right)\left(c+1\right)}}{2}+\frac{\sqrt{\left(c+1\right)\left(a+1\right)}}{2}\right]^2.$$](http://latex.artofproblemsolving.com/8/3/0/830a239d407ecde3df844d7d989e273bb41209bb.png)
When does equality hold?
and 
which are such that 
is divisible by 
.
?
is divisible by 
be a positive integer. Anu has an infinite collection of notes with exactly 
copies of a note worth 
rupees, for every integer 
.
is called payable if Anu can pay exactly 
rupees by using some collection of her notes. Prove that if there is a payable number, there are infinitely many payable numbers.
.
is a parallelogram. 
and 
.
and 
is 
.
.
is an equilateral triangle such thats perimeter is equal to the area of its inscribed circle. Find the possible values for the radius of said circle.
is perpendicular to 
,
is perpendicular to 
and 
.
,
is on side 
.
is the foot of the perpendicular descending from 
onto 
.
and 
.
.
are taken on sides 
,
is parallel to 
,
,
.
and quadrilateral 
have the same perimeter, find 
.
and the height from 
.
are the midpoints of sides 
,
inside the triangle such that 
.
.
of a quadrilateral 
.
,
,
.
.
is inscribed in a circle. Let 
be a point on the line segment 
such that 
,
.
intersects the circumcircle of 
,
and 
.
,![\[ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 46 \qquad
\textbf{(D)}\ 66 \qquad
\textbf{(E)}\ 74
\]](http://latex.artofproblemsolving.com/8/2/f/82ffc15072ca99f6cd1abc312781eb4827817356.png)
and 
,
and 
.
.
Y by Funcshun840
Let 
be an acute triangle. Let 
be its incenter, 
and 
such that 
, 
, 
are collinear. Let 
such that 
. 
is defined similarly. Let 
such that 
. Prove that 
.
Proposed by Pavel Ciurea
be an acute triangle. Let 
be its incenter, 
and 
such that 
, 
, 
are collinear. Let 
such that 
. 
is defined similarly. Let 
such that 
. Prove that 
.Proposed by Pavel Ciurea
This post has been edited 1 time. Last edited by Marius_Avion_De_Vanatoare, Jun 14, 2025, 7:44 PM
since 
.
be the antipode of 
and 
,
is parallel to 
.
(the last step follows from rotating each line by 
degrees). A simple config analysis allows us to conclude that 
is 
,
