The document discusses limits and their applications. It defines a limit as our best prediction of a value we cannot directly observe. Limits are useful for scenarios in mathematics that involve division by zero or going to infinity. Real-world objects move continuously between positions, so limits allow us to make accurate predictions. The document provides examples of using limits to predict the position of a soccer ball in a video and discusses how limits apply in areas like physics, engineering, and natural phenomena.

1. Limits and Their
Applications
MR. HASSAN HLEIHEL
HAROUN EL MIR

2. What is a limit?
Why do we need limits?
Explanations
Examples

3. What is a limit?
Our best prediction of a point
we didn’t observe.

4. How do we make a prediction?
Zoom into the neighboring points. If our prediction is always in-between
neighboring points, no matter how much we zoom, that’s
our estimate.

5. Why do we need limits?
 Math has “black hole” scenarios (dividing by
zero, going to infinity), and limits give us a
reasonable estimate.

6. How do we know we’re right?
 We don’t. Our prediction, the limit, isn’t required
to match reality. But for most natural
phenomena, it sure seems to.

7. Limits let us ask “What if?”. If we can directly observe
a function at a value (like x=0, or x growing infinitely),
we don’t need a prediction. The limit wonders, “If
you can see everything except a single value, what
do you think is there?”.
When our prediction is consistent and improves the
closer we look, we feel confident in it. And if the
function behaves smoothly, like most real-world
functions do, the limit is where the missing point must
be.

8. Predicting A Soccer Ball
Conclusion:
Real-world objects don’t
teleport; they move through
intermediate positions along
their path from A to B. Our
prediction is “At 4:00, the ball
was between its position at
3:59 and 4:01″. Not bad.
Limits are a strategy for
making confident
predictions.

9. What things, in the real world, do
we want an accurate prediction
for but can’t easily measure?

10. Math English Human English
means
When we “strongly predict” that f(c) =
L, we mean
for all real ε > 0
for any error margin we want (+/- .1
meters)
there exists a real δ > 0 there is a zoom level (+/- .1 seconds)
such that for all x with 0 < |x − c| < δ,
where the prediction stays accurate
we have |f(x) − L| < ε
to within the error margin
•The zoom level (delta, δ) is the function input, i.e. the time in the video
•The error margin (epsilon, ε) is the most the function output (the ball’s position) can differ from our prediction
throughout the entire zoom level
•The absolute value condition (0 < |x − c| < δ) means positive and negative ways must work, and we’re skipping the
black hole itself (when |x – c| = 0).

11. Could we have multiple predictions? Imagine we
predicted L1 and L2 for f(c). There’s some
difference between them (call it .1), therefore
there’s some error margin (.01) that would reveal
the more accurate one. Every function output in
the range can’t be within .01 of both predictions.
We either have a single, infinitely-accurate
prediction, or we don’t.
Ask for left and right hand limits
(always)

12. For example: Prove the limit at x=2 exists for
The first check: do we even need a limit?
•Assume x is anywhere except 2

13. For any accuracy threshold (ε), we need to find the
“zoom range” (δ) where we stay within the given
accuracy. For example, can we keep the estimate
between +/- 1.0?
In other words, x must stay within 0.5 of 2 to maintain the initial accuracy requirement of 1.0. So,
when x is between 1.5 and 2.5, f(x) goes from f(1.5) = 4 to and f(2.5) = 6, staying +/- 1.0 from our
predicted value of 5.

14. Examples:
if we are 10 feet from the wall and walk half the distance each time, will you ever reach the wall?
without the help of calculus most students would be pretty stumped about this question..applying
calculus it is obvious that you would eventually hit the wall
a speedometer in a car is constantly calculating a limit where time goes to zero...if you
calculated velocity over a certain range of time it would be an average velocity not
instantaneous. since v=d/t, in order to make t = 0 one would need to use a limit
in engineering we use it to calculate areas of complex shapes where basic geometry fails

15. Only so many cars can be moving on a given road. Traffic flow limits.
These limits are represented by stops. If you remove one, the flow may increase till the next limit is
found (a new stop).
Only so many messages can go through a given network. Another traffic flow limit.
There is a limit to how much food one can eat, or how little.
Limit to how fast you can go on the road. Many limits like this are imposed by regulations.

16. A sky diver or any object reaching terminal velocity
when falling.
When F air resistance = F gravity
Another is predator-prey systems.
An example would be a fox population hunting
rabbits.
Depending on the number of foxes and rabbits and
their death rates and hunting rates, after a long time,
or the limit of the populations can reach stable
levels, can be cyclic, or both die out and eventually
reach a limit of zero.
Limits are everywhere in nature.

18. The reading of your
speedometer (e.g.,
85 km/h) is a limit in
the real world. Your
speed is changing
continuously during
time, and the only
"solid", i.e., "limitless"
data you have is
that it took you
exactly 2 hours to
drive the 150 km
a guitar string
vibrating. It starts out
with large
displacement and
eventually becomes
still.