Why We Study Limits In Calculus
Audience groans
Today we will be learning about limits, but before we learn
about limits, we need to know why we study them. In order to do that, we have
to tell the story of Zero.
This is the story of Zero.
Zero was born slightly different. He simply wasn’t like the
other kids. He wasn’t like the good positive numbers, like number one, tall
enough to grab a book from the top shelf without a stepladder. He wasn’t like
2, who’s graceful black hair fell down perfectly straight along her back. He
couldn’t crack jokes like 3 or even snap his fingers like 4. He wasn’t as fast
or strong as 5, nor as agile or flexible as 6. He couldn’t even understand some
of the words 7 was already using, and 8 could do a headstand. 9 could do
arithmetic faster than Zero could write down the problem.
The bus had ten seats and Zero would place his backpack on
the seat beside him as soon as he sat down.
The cafeteria had triangular tables for 3 people each, and
Zero would always have an entire table for himself.
When Zero wanted to play tag with everyone else, the game
ended. Without him being tagged at all.
After a few weeks went by, Zero sat down, despondent, and
cried. Not a sound was made.
Zero’s house aptly called “Property of Zero (Additive).” He
was never recognized because when you add zero to another number you see Zero
fade away, as if he never existed. He even wanted the mean negative numbers to
notice him, but despite his additions to their conversations, it was like as if
he never existed at all.
Finally, one night, in bed, Zero tried everything. He added
zero to himself, multiplied zero to himself, subtracted zero to himself. Then
when everything was finally over, he divided zero by himself. The mirrors
around him that he checked all showed the worst had happened… Calculator error
signs. He collapsed into his bed, hoping for it all to be over. It was over
now, right?
Wrong.
When he woke up, sweat on his pillow, he frantically scanned
himself in the mirror to see if he had changed. He hadn’t.
However, unbeknownst to him, when he was walking outside his
house, the insignia was no longer a bronze cross, but rather a bronze X, and
the nameplate read “Property of Zero. (Multiplicative)”
Along the way he bumps into 5, and now, instead of 5 simply
pushing him away, upon contact, 5 collapses onto the ground, writhing in pain,
until he himself turns into a zero. At first Zero watches this happen in
terror, then a grin spreads across his face as wide as the minor axis that
makes up his elliptical face. Today he would no longer be Zero.
On the bus, he grabs a hold of 7, creating another zero. The
rest of the students shriek and panic and hide from zero. At school, Zero shows
no mercy. He chases after everyone, finally able to have his turn to be “It.”
Finally, Zero approaches his last target. Eyes shining in
bloodlust, Zero lashes out at Infinity. Infinity was a stellar student in every
aspect. He was placed into the gifted and talented program, but constantly felt
bored in class. No one could understand his ideas; even his teachers had
trouble understanding even the language he used. Zero lashes out his crossed
arms, but Infinity holds him back. Infinity pushes Zero to the ground,
struggling to stop him, but during the struggle, the winner was Indeterminate.
Infinity rose out, then called out to Zero “Will you ever stop this? I
understand how you feel!” Zero called back “How could you ever understand how I
feel, you’ve never been treated like nothing!”
“You don’t think I am treated like nothing? I’m the biggest
number there is, leaps and bounds ahead of everyone. No one understands me at
all. The teachers don’t know what to do with me and neither does anyone else.”
“Then why are we having this conversation?”
“Because perhaps maybe we have both felt the same thing for
a long time. Maybe it’s time to stop this, Zero, and help the numbers
understand us.”
So, Infinity helped rebuilt the other numbers.
When you study calculus, remember that you are studying the
relationships between numbers. The idea that we can approach any number, even
ones that are nothing or everything, and understand them: that is what a limit
is.
Limit as x approaches infinity of 1 over x, 2 over x, 3 over
x, 4 over x, and on... they all approach zero. The people who we never see, who
silently and secretly support numbers like 10 and 100 and 1000, they must be
appreciated and understood. They deserve that much for the work they do. That
is why we study limits. Thank you for listening class. If I ever come back next time, I'll be teaching you about the rebellious student Derivative.
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