,
,
,
,
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 
such that 
and 
,
.
be fixed points on this order on a line. Let 
be a variable circle through 
and 
and suppose it meets the perpendicular bisector of 
at the points 
and 
.
and 
be the other points of intersection of 
and 
with 
passes through a fixed point independent of 
and 
.
be a positive integer and let 
be distinct integers in the set 
such that 
for 
.
and 
are numbers that work, but 
and 
don't work.)
Y by nguyendangkhoa17112003, Purple_Planet, Adventure10, megarnie, Mango247
In the plane let 
be a circle, 
a line tangent to the circle 
and 
a point on 
. Find the locus of all points 
with the following property: there exists two points 
on such that is the midpoint of 
and is the inscribed circle of triangle 
.
be a circle, 
a line tangent to the circle 
and 
a point on 
. Find the locus of all points 
with the following property: there exists two points 
on such that is the midpoint of 
and is the inscribed circle of triangle 
.
at 
be the center of 
be one of points that has the problems property. Let 
be the vertices of triangle which has the problems property. Let 
be on 
.
such that be tangent to 
externally . and let 
intersects 
is center of homothety which convert 
are collinear and since 
are constant , so the locus of 
be 
,
is a diameter of 
and 
be 
,
.
are fixed, 
,
such that 
![[asy]
size(8cm);
defaultpen(fontsize(10pt));
pair P = dir(120), Q = dir(210), R = dir(330), I = incenter(P,Q,R), T = foot(I,Q,R), M = (Q+R)/2, S = Q+R-T, Tt = 2I-T;
dot("$P$", P, dir(45));
dot("$Q$", Q, dir(270));
dot("$R$", R, dir(270));
dot("$T$", T, dir(270));
dot("$M$", M, dir(270));
dot("$S$", S, dir(270));
dot("$T'$", Tt, dir(240));
draw(Tt--Tt+dir(S--Tt)*1.2, red, Arrow);
draw(Q--P--R, blue);
draw(2Q-T--2R-S, Arrows);
draw(incircle(P,Q,R));
draw(T--Tt--S, blue+dashed);
[/asy]](http://latex.artofproblemsolving.com/8/7/e/87ef6514718b365f8774eaecf3f51923ac0bb77e.png)
be the point diametrically opposite 
)
.
,
and 
such that 
and 
of 
,
be the reflection of 
.
is the excenter of 
(proof by homothety). Then, the midpoint of QR is the midpoint of 
which is 
fail since 
,
is still the excenter of 
which is not equal to the midpoint of 
since 
,
be center of 
and 
be reflection of 
over 
be antipode of 
.
where 
such that 
.
and 
We claim that 
beyond 
Since 
is the incircle of 
we know 
at 
noting that 
