,
,
,
,
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 the circumcircle of triangle 
is the midpoint of 
be the second intersection of 
and 
,
be a transversal of 
be intersections of 
with 
be a tangent line of 
is cyclic
be positive integers such that 
and 
.
.
and 
Prove that
:o
so 
and since 
is just 
therefore 
so 
such that![\[
\sqrt{x - 2011} + \sqrt{2011 - x} + 10
\]](http://latex.artofproblemsolving.com/4/8/b/48b82d13b733947420a0d763c6af74636770835a.png)
is an integer.
-
?
students want to share 
pencils. If every student gets at least one pencil, how many ways are there to distribute the pencils?
.
Compute the minimum possible value of 
Y by Adventure10, Mango247
Given an acute-angled triangle 
, let points 
be located as follows: 
is the point where altitude from 
on 
meets the outwards-facing semicircle on as diameter. Points 
are located similarly.
Prove that![$A[BCA']^2 + A[CAB']^2 + A[ABC']^2 = A[ABC]^2$](//latex.artofproblemsolving.com/8/d/9/8d9857d4abd79b53da9a0b743017c3bfd6f6636a.png)
where ![$A[ABC]$](//latex.artofproblemsolving.com/5/7/0/570d1f902c11da5be088156e7edf95da0e8682e4.png)
is the area of triangle .
, let points 
be located as follows: 
is the point where altitude from 
on 
meets the outwards-facing semicircle on as diameter. Points 
are located similarly.Prove that
![$A[BCA']^2 + A[CAB']^2 + A[ABC']^2 = A[ABC]^2$](http://latex.artofproblemsolving.com/8/d/9/8d9857d4abd79b53da9a0b743017c3bfd6f6636a.png)
where ![$A[ABC]$](http://latex.artofproblemsolving.com/5/7/0/570d1f902c11da5be088156e7edf95da0e8682e4.png)
is the area of triangle .This post has been edited 1 time. Last edited by Rushil, Oct 4, 2005, 5:43 AM
,
,![\[ (\dfrac{\triangle{BA'C}}{\triangle{BAC}})^2=\dfrac{BD\cdot CD}{AD^2} \]](http://latex.artofproblemsolving.com/9/e/6/9e6bef00e34848d3d8ac4ed93e136ad282e66b42.png)
,
,![\[ \sum \cot{B}\cot{C}=1 \]](http://latex.artofproblemsolving.com/4/1/9/4191ef04e467cd589b224d13aedd496fc1f84adf.png)
![$[ABC] = \Delta, [HBC] = \Delta_1, [HCA] = \Delta_2, [HAB] = \Delta_3$](http://latex.artofproblemsolving.com/c/c/c/cccabf746c5c53311c72f58140ccca2325dc4b51.png)
![$[A'BC]^2 = [ABC][HBC] = \Delta \Delta_1$](http://latex.artofproblemsolving.com/7/1/c/71c1daea2122f0c2b00591d2aa71e5f656964393.png)
![$[B'CA]^2 = \Delta \Delta_2 \text{ and } [C'AB]^2 = \Delta \Delta_3$](http://latex.artofproblemsolving.com/6/1/e/61e087b25fe09d8440597f7f838e8ef3bbeb9bcb.png)
![$[A'BC]^2 + [B'CA]^2 + [C'AB]^2 = \Delta (\Delta_1 + \Delta_2 + \Delta_3) = \Delta \cdot \Delta = [ABC]^2$](http://latex.artofproblemsolving.com/7/2/0/720526b9a93ceb28f410240fcca4038219678597.png)
,
meet 
at 
respectively.
and 
are two similar triangles.
and 
are two similar triangles.
and 
are two similar triangles.
,
,
