,
,
,
,
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 
?
\end{table}
has how many digits?
is a quadratic polynomial such that 
and 
.
.
have the digit 
and are divisible by 
?
![\[(2+0+2+4)\left((2\cdot2\cdot4)^2-\dfrac{2^2+0^2+2^2+4^2}{2+0+2+4}\right)=2024.\]](http://latex.artofproblemsolving.com/0/0/7/0074cd2c9c2f8327db52e113ef2beefe98ceca90.png)
![\[(2+0+2+4)(2\cdot2\cdot4)^2-\left(2^2+0^2+2^2+4^2\right)=2024.\]](http://latex.artofproblemsolving.com/d/5/4/d54152fae9e6e259dc464dbef3ed39272eb9e599.png)
Y by saksham1729, RAMUGAUSS, Adventure10, Rounak_iitr
Let 
be the diameter of a circle 
and let 
be a point on different from 
and 
. Let 
be the foot of perpendicular from on to .Let 
be a point on the segment 
such that 
is equal to the semi perimeter of 
.Show that the excircle of opposite is tangent to .
be the diameter of a circle 
and let 
be a point on different from 
and 
. Let 
be the foot of perpendicular from on to .Let 
be a point on the segment 
such that 
is equal to the semi perimeter of 
.Show that the excircle of opposite is tangent to .This post has been edited 2 times. Last edited by div5252, Jan 20, 2019, 5:54 PM
as the circle tangent to 
,
and 
.
,
.
.
,
.
.
be the midpoint of 
the semiperimeter of 
and 
.
,
-![\[AB\cdot AD = AC^2 \implies AB = \frac{s^2}{k}\]](http://latex.artofproblemsolving.com/f/b/f/fbfdc2c25cbe2b2c87fa0bd6d9f35a8af3e44eaa.png)
![\[AM = \frac{s^2}{2k} \implies M = \left(\frac{s^2}{2k} - k, 0\right)\]](http://latex.artofproblemsolving.com/5/a/6/5a6b72a7c88db9b21d12c1071aae0ec483b1f911.png)
Let 
be the 
,
.
So, done.
be 
and 
be tangent to 
,
.
or 
,
be A-excircle of 
)
at 
,
and has center at 
,
is a square and , 
,
and recall that 
, forces, 
.
be the points where 
.
Let 
be denoted by 
.
,
,
.
as claimed.
and circle 
excircle of 
and let 
at 
.
.
.
.
as 
.
.
.
are collinear. But how will you show that there is not other common point for the two circles other than 
be the point where the 
(say 
,
centered at 
be the inversion respect to 
and 
;
and 
of 
and 
.
,
.
we get that 
as 
as 
where 
are collinear and 
and 
are contact points of excircle with 
.
maps to 
with is tangent to excircle.
opposite 
has length 
,
be the radius of the excircle and 
.
be the center of the excircle. We need to prove that 
.
,
hence we need to prove that 
.
also, 
Then,
,
,
which is centered about the circle 
.
,
.
be 
is tangent to 
I.e. to 
and 
.
which is clearly tangent to 
are orthogonal (so it stays put) and that the line CD and 
.
.
at 
and touch 
are collinear and it's also well known that 
and also we know 
equals semi perimeter of 
