Eric C. answered • 12/16/15
Tutor
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Engineer, Surfer Dude, Football Player, USC Alum, Math Aficionado
I always liked to think of limits as a way of cheating algebra. We all know in algebra that you can't divide by 0, but when evaluating a limit as h approaches 0, you're essentially saying, "I'm going to get so close to zero that the value might as well be zero, but I'm not going to actually hit 0 so the rules I learned in algebra don't apply."
When using the limit definition of the derivative, you actually divide by 0 and say it's okay because you're not "actually at 0." Take a look:
Say you want to compute the derivative of x^2 using the limit definition:
lim [f(x+h) - f(x)]/h
h->0
This becomes
[(x+h)^2 - x^2)]/h
[x^2 + 2xh + h^2 - x^2]/h
(2xh + h^2)/h
h(2x + h)/h
It's right here that you do the classic trick of canceling the h's, then plugging in 0 for h afterward.
lim h(2x + h)/h
h->0
lim 2x + h = 2x
h->0
If you plug in 0 for h, though, you should not be allowed to cancel them. Just because the h's are invisible now doesn't mean they aren't there. Algebra would say this is wrong.
But with a limit, you don't actually "hit" 0. You might be at 0.000000000000000001 but that's still not 0. So when you do your canceling, it's with non-zero numbers, but we plug in 0 to make our calculations easier because it's "close enough that it might as well be." By adding the term "lim" in front of a function, you're essentially allowing yourself to cheat at math, but in a physical reality it's okay.
Derivatives compute things that happen "in an instant." In math, the definition of an "instant" is something that occurs during which the change in time is 0. You can be asked "what's your instantaneous velocity" or "what's the change in volume at this instant" or "how much money was I making at this very instant".
To find a rate of change of something that happens where the change in time is 0 using classical algebra is impossible, because a rate is always something/time, and you can't divide by 0. But more importantly, a change of time of 0 doesn't represent anything that happens in reality. A football player may make a very fast change in direction that looks like a sharp cut, but no matter how fast the cut happens, if you zoom in close enough on his trajectory [i.e. when the change in time is small enough], you will see a smooth curve which can be evaluated. This is what is meant by a limit; zoom in close enough on your time so that that change is essentially 0, but it never actually is 0.
So even though algebraically, limits are means of cheating, they're actually more representative of reality than the rules of algebra dictate, because even when you get down to molecules and particles, eventually the infinitesimally small stops getting smaller. That's why they're used so much in engineering.
Some examples:
If you modeled the resolution of your vision as some function and took its limit as the resolution approaches 0, you would get a real number. This number is the same resolution that computer printers use, i.e., how many "small dots" are placed in a square inch to make it seem like a solid photo. If you took the limit as the change in distance approached 0 on your computer printout, you'd eventually see dots. This is also why HD televisions come in 1080p and not infinity pixels. At 1080p, the difference in resolution from what your eyes can receive and what the television can produce is close enough to 0 not to matter anymore. Just like when we said "0.000000000001 is close enough to 0 that it might as well be 0", for a TV manufacturer, the same logic applies. This saves in cost on pixels per TV and it gets the same result for the consumer.
Same thing with audio files. An mp3 is a collection of discrete sound bytes that are played in a sequence, but there are so many of them playing per second that it sounds like a smooth noise coming out of the computer. This is another engineering problem where limits come into play; what's the change in time required to make the discrete sound bytes seem smooth in a human ear? Take a limit of your model and find out what it approaches, then make that your frame rate.
Hope this helps.
Eric C.
tutor
Let's take the example of flipping a coin. We'll say one coin has a heads on one side and a tails on the other.
You want to know the probability of flipping this coin and never landing on heads.
Intuitively you know this is pretty much impossible. But how can it be described using math??
The probability P of flipping a two-sided coin and landing on tails is 1/2. The probability of flipping it twice and landing on tails twice is 1/4. Flipping it three times and landing on tails three times in a row is 1/8, etc.
The probability of flipping a coin n times and landing on tails all n times is:
P(n) = (1/2)^n
You want to know the probability of never landing on heads. What that says is that n will be infinity. What happens to your probability P as n approaches infinity?
lim n-> ∞ (P(n)) = (1/2)^∞ = 0
Your probability "approaches" 0 as n "approaches" infinity. Using the logic and definitions I described above, you'll know that 1/2 gets really really close to zero, but it never quite "hits" it, because no matter how many times you multiply 1/2 by itself, there will always ALWAYS be a 1 in the numerator, meaning it'll be positive and NOT zero. But, for the statistician's sake, it's best to assume the probability is actually 0, so he's not likely to bet on this scenario happening, even though it is technically possible.
What about if we had a coin where both sides were tails? What's the probability of flipping this coin and never landing on heads?
The probability of flipping THIS coin n times and never landing on heads is:
P(n) = (1)^n
If we want to know what happens if we flip it infinite times, we take the limit:
lim n-> ∞ P(n) = (1)^∞ = 1
The probability in this case is 100% never landing on heads, which we knew intuitively, but now proved mathematically.
Probably the most famous example of where algebra breaks down and limits better describe reality is the math student trying to get out of coming up to the whiteboard by arguing it's physically impossible to get there.
He claims that in order to go from his desk (point D) to the whiteboard (point W) he'll have to traverse the distance DW. but in order to reach DW, he must first cross some point M midway between D and W. So DM = 1/2 DW. But in order to reach M, he first has to reach a different midpoint N between D and M, so DN = 1/2 DM. He continues on like this and claims he'll never reach the whiteboard because he'll always have some positive distance to travel.
This limit looks very similar to the probability limit of a fair coin. If he has to travel an infinite number of 1/2 distances, then all in all he'll travel:
DW*∑ (1/2)^n, n=1 to ∞
This is a geometric series which evaluates to:
DW* [lim n-> ∞ (1-(1/2)^n)/(1-1/2)]
The (1/2)^n term goes to 0 as n goes to infinity. You're left with 1/ (1 - 1/2) = 2
Using limits, the professor showed the student that he has to travel a non-infinite distance to get to the whiteboard.
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12/17/15
Justin J.
Thanks for your comment but this just explain why limits have born.I would like to know how limits are used in reality.It may be better if you could show how to solve limits using the example you have already mentioned(resolution example).
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12/17/15
Justin J.
12/17/15