Math Limits; Algebraic and L'Hopital's, and Black Holes:
CHRISTIAN HEADS UP:
1. All math hubs have a comment about God in there last paragraphs. In this hub these are at paragraphs 19, 20, 21, and 22.
LIMITS--A FOUNDATIONAL CONCEPT
2. The concept of limits is foundational to differential calculus. Differential calculus could not exist without the concept of limits. In fact it is defined by the limit. Furthermore, Thomas and Finney said in their fine text, Calculus, 9th ed, page 333 concerning "the Fundamental Theorem of Integral Calculus":
3. "This conclusion is beautiful, powerful, deep, and surprising, and Eq. (2) may well be the most important equation in mathematics. It says that the differential equation dF/dx = f has a solution for every continuous function f. It says that every continuous function f is the derivative of some other function, namely ∫ax f(t) dt . It says that every continuous function has an antiderivative. And it says that the processes of integration and differentiation are inverses of one another."
4. And it says to me that Integral calculus also would not exist without the concept of limit in mathematics. None of the examples of limits in this hub go to plus or minus infinity( ∞ ), but if we include these limits, then much of mathematics as we know it would not exist; certainly not analysis. So without limits, calculus( which is actually an introduction to analysis) and analysis would be no more. This would put most engineers out of business. I mean, really, what would electrical or electronics engineers do without sin(θ)/θ as θ→0? Drink lots of coffee, probably. There will not be much else to do without the concept of Limits.
BASIC ARITHMETIC, YOU CAN'T DO MUCH WITHOUT IT :
5. Actually, you can't do anything without it, not in math. If you get nothing else from this hub, I hope you learn that basic arithmetic is foundational for doing more advanced math. You cannot solve anything without it. In this hub I have used some of the most basic math taught in school. For example, if X is any real number then X times 1 is X; X divided by X is 1, and X = X. You can't get much more basic than that, and yet this stuff is used to solve the limits in this hub.
SIN(θ) / θ = 1, AS θ→0. VERY IMPORTANT !
6. Sin X divided by X equals one as X approaches zero. This is used very much in physics and engineering, especially electrical engineering. The fact that this function equals one as the angle approaches 0 is proven in calculus, and this fact is also used to solve many limits as in some of the examples in this hub. I've seen some pretty ingenious proofs of this famous limit, but the most common is one done with the squeeze theorem.
DIVISION BY ZERO IS UNACCEPTABLE:
7. At L1 if we plug in 0 for X, then both the numerator and denominator go to zero. We can't divide by zero; therefore we manipulate this function until we can get rid of zero in the denominator. Sometimes it is there to stay and the function goes to + or - infinity(∞). At L2 we use this trig ID( trigonometric identity) to convert the tangent function into a sine function after we multiply the numerator and denominator of L1 by cosine X at L3, but we still have 0 / 0 in the 2nd term of L3 when we plug in 0 for X.
CONVERT SIN X INTO (SIN X) / X TO GET ONE:
8. Your basic math says you can multiply the numerator and denominator of any fraction by the same number and the value of the fraction does not change; therefore, that is what we do at L4. It goes without saying that X ≠ 0. After doing the manipulation at L4 we have L5 in a form that reduces to the limit, which is 0.5. At L6 we introduce L'Hopital's rule. This is pronounced " low pee tall". Various textbooks spell his name differently, as both ways I have at L6 with an "s", and here without the "s". I prefer without the "s" since it coincides more with the proper pronunciation of his name. L'Hopital's rule is as powerful in solving limits as the quadratic formula is in solving for the zeros of a quadratic; however, there are certain conditions that must be met, and one of those conditions is the limit function must be in the form of 0 / 0, or if you can transform it into that form then L'Hopital can be used.
DO NOT USE THE QUOTIENT RULE---DIFFERENTIATE SEPARATELY:
9. Once the condition is met, L'Hopital says the original limit function equals the same limit function but with the numerator and denominator differentiated separately. YOU DO NOT USE THE QUOTIENT RULE OF DIFFERENTIATION IN THIS CASE. You differentiate the numerator and then differentiate the denominator, and then take the limit and see if it works( gives you an answer other than 0 / 0). If you do get 0 / 0 then you can use L'Hopital again. The condition is met at L7, so we execute L'Hopital's rule at L8, and find that we can take the limit of X--->0 and get an answer of 0.5 at L9. L10 just explains the notation used.
SUBSTITUTION--A POWERFUL CONCEPT IN MATHEMATICS:
10. We learn about the technique of substitution in algebra, and from there on we use it to solve equations. Even techniques as Laplace transforms, and Fourier transforms are, in my opinion, a type of substituting to make equations easier to solve. So that is what we do. We substitute L12 into L11 and get L13, which we recognize as the difference of two terms in the denominator; therefore, we can cancel out the factors and get L14, which is a very easy limit to solve. Don't forget to convert the limit back to its original variable; otherwise, you will get the wrong answer. We did this at L14, and 15.
L'HOPITAL TO THE RESCUE:
11. L16, 17, and 18 shows how L'Hopital is used on L11, but L19 is a very good example of how L'Hopital can make life easier. We simply use the Power Law of Differentiation to easily solve L19 at L20. L21 is solved by using a basic trig ID, and recognizing the famous Sine X over X limit.
USE PREVIOUS LIMITS:
12. At L26 if we can get it in the form of L21, then we know the value of L21 is one at L25. So we multiply the numerator and denominator by L27, at L28 and get L29, which has at least the numerator in the form we want. In this case the laws of limits allow us to take the limit of the numerator and denominator separately at L29, and we know that the limit of the numerator is 1, but we still have to deal with the denominator. L31 shows that we can factor out I / 4th and get the denominator in the form we need. The laws of basic arithmetic tell us that a complex fraction as the denominator of L30 puts the 4 in the numerator, which leaves us (tan 12X) /(12X), which has a limit of 1 as X approaches 0; therefore, our answer is 1 / 4. Now L33 is a perfect limit with which to use L'Hopital's rule. This would be a tough limit to solve, at least for me, without L'Hopital. If zero is substituted for X in L33 then we have 0 / 0, which meets the condition for using L'Hopital.
THE LIMIT OF AN EXPONENTIAL FUNCTION:
13. At L35 we differentiate the numerator and denominator separately and get the correct value of 0.5878. At L38 we decompose this function into its factors, and then get them in the form we need at L40, and get the answer at L41.
DON'T BE LAZY, BUT SAVE THE TREES WHEN POSSIBLE:
14. You'll notice I stupidly used X---->0 when I should have used t---->0 or I should have put X in place of t for the variable. Just read t---->0 for the limit. It is not that I'm too lazy to change things, but I have to use a new piece of paper to do it. You will notice I'm always making mistakes and if I grabbed a new piece of paper every time, then we would have no forests left. . . . What? Well, no I don't exaggerate---of course not---why would you say that?
15. Anyway, here's my thinking. A sheet of copy paper weighs about 9.022 X 10-3 pounds. Let us say that everyone in the United States makes a conscience effort to not waste a sheet of paper a day; this is extremely conservative considering students, newspapers, and all other variables. So that works out to be 300,000,000 sheets of paper a day let alone what the whole world would save if they did the same thing. That would be a stack of paper 66,000 feet high( 12.5 miles) and it would weigh 2.9766 million pounds( 1,488.3 tons)! In one year the stack would be 4,562.5 miles high, and weigh 543,229.5 tons. So, I have a thing about wasting paper.
AN AMAZING LIMIT:
16. At L38 we want to get it in the form of sinθ / θ, which we can do by multiplying the numerator and denominator by 7 at L40, but first we had to decompose the function into its factors at L39. One times any number is that same number; therefore, we do this to both factors. We isolate the famous sine limit and realize it represents one; therefore, we essentially have 7 times 7 equals 49.
17. There are many interesting limits, but this one caught my attention because of how big it could get. At L42( that is supposed to be t--->0, NOT x--->0) if you plug in zero for t you will get 0 / 0. That is a mathematical fact. Yet when we find the behavior of the limit it gets huge. I chose an exponent of 153 because that is how many fish they caught when Jesus told them to give it another try but in this way( John 21:1--11, verse 11 says 153 great fishes). I chose 49 because that is the square of 7, which is the number of "spiritual perfection" in Scripture( "Number in Scripture--Its Supernatural Design and Spiritual Significance" by E. W. Bullinger). The number 3.98 X 10258 is a huge number. If you filled the entire universe with sand, it would only contain 1091 grains.
17.1 For more techniques on solving math limits go to hub#12.14( e, The base of . . . .)
YOU WILL LOVE THIS BOOK:
18. "Number in Scripture" by E.W. Bullinger is fascinating. In all my hubs I try to choose significant numbers that have meaning in scripture. For example, the number 223,776 in the hub, Proof of the Bible, pt 1 has a whole bunch of Scriptural meaning; however, it must be factored to extract all the Biblical meanings. Bullinger's book gave me the ability to give my numbers meaning when I have the option of choosing a number. It is not available at Amazon at the time of this writing( 3-21-12) but I linked you to the publisher. It is only 16 bucks--a great deal! I went to the book itself but the link only goes to the publisher; however, just type in the title and you will get the book. The date now is 4-7-12 Saturday, and Amazon has some copies; however, I'll leave the link to the publisher in case Amazon runs out.
Limits are taught in pre-calculus, and calculus
GOD KNOWS HOW TO CREATE REAL LIFE LIMITS:
19. I suppose there are numerous examples of real life limits in God's creation, but the one nearly everyone knows about is Black Holes. They are about as cool as you can get. Doctor Melkor (link takes you to his hub) does a very thorough and lucid job of explaining them to us. I think it can be said with confidence that a Black Hole takes Newton's equation of gravitation and converts it into a limit: Limit( r---->0): (GM1Mblack hole) / r2. Personally I do not believe the radius of a Black Hole( collapsed star) gets to zero but rather it stops collapsing eventually; however, since I have not studied General Relativity, the mathematics that describe Black Holes, what do I know? This is a case in which the limit approaches infinity as the radius approaches zero. If I were to choose just two items--there are so many more--that display God's great power, they would be black holes and quasars. They are both examples of how God assigns limits to His creation, but those limits are enormously impressive.
BLACK HOLES AND GOD'S OMNIPOTENCE:
20. I try to catch those educational shows whenever possible. I've seen several that claimed that if you were at the event horizon of a black hole then the gravitational force upon your feet would be so much greater than your head that you would be forged into something similar to a piece of spaghetti. What do ya say we put some numbers to that. Stars that form into black holes are much larger than our sun, but I want to use our sun for our example because we are so familiar with it. If our sun was a black hole then its Schwarzschild radius, the event horizon, would be 2,953 meters( 1.835 miles--less than 2 miles)! If we consider 3 pounds of your feet then the gravitational force on those 3 pounds would be 4.65479729 trillion tons, and on your brain, which is also about 3 pounds, it would be 4.64903769 trillion tons. That gives us a difference of 5.75960206 billion tons. So you would have nearly 6 billion more tons pulling on your feet than your head! It would be progressively that way throughout your body. Yes, that would most definitely forge you into spaghetti; however, it is one way to reduce the waistline, and narrow those hips. When I encounter numbers like this I do not say, Wow, gravity is awesome, I say "Wow, God is awesome." God created--through Jesus Christ--matter, gravity, forces, energy, and . . . well, everything. These are God's numbers, not gravity's. These numbers glorify God and nothing or no one else.
Limits are given structure in calculus
WHY DOES THIS GRAVITATIONAL LIMIT GLORIFY GOD?
21. Because God, by Jesus Christ, is the Creator. It is as simple as that. You evolutionists cannot answer any of my questions. You cannot tell me the origin anything: not gravity, space, time, the electromagnetic force( emf), the strong force, color force, weak force, energy, photons, gluons, first membrane of a cell, information stored in DNA, matter, electrons, protons, neutrinos, neutrons, potential energy, kinetic energy, Pauli Exclusion Principle, Heisenberg's uncertainty principle, Planck's constant, Fine Structure constant, permittivity, permeability, moving electrons making magnetic fields, . . . not a damn thing. All these things must be "naturally" explained if God is not considered to be the Creator. Do you actually expect anyone to take the valuable time that God has graciously given to them, and inanely spend it coming up with fatuous conjectures( don't you dare call them theories) of how a book came into existence through random processes when everyone knows it has an author, a publisher who published it, a logger who got the wood, and a manufacturer who converted wood to paper? Do you think I'm going to waste my time coming up with ridiculous postulates of how my computer came into existence by random processes? Or how a tornado can build a skyscraper complete with plumbing and wiring? If you can't see God as the magnificent Creator that He is, then I suggest you bend those stubborn and insolent knees and ask Him to open your inanimate blind eyes, because if you don't, then you will have eternity to regret and remorse that you had not. Don't be such a fool! In your entire life have you known anything inorganic that has built itself, repaired itself, or had the slightest tendency to violate entropy( going from order to disorder). All elements are inorganic, including carbon. What makes carbon organic is how other elements connect to it, including carbon itself. Those connections are precise and designed, and that is done by God. There may be debate on just how God does this as I explained at paragraph 21 of Transforming the Fibonacci Sequence, but the fact that God by Jesus Christ is the ultimate mechanism that keeps the universe working and together is clear. I would be interested in your best argumentation that will contradict this fact.
DON'T BE A FOOL; CONSIDER YOUR ETERNAL DESTINY:
22. The Bible describes who God is. In addition to being a God of love, He is also a God of judgment. Our opinions about God are useless. We must rely on what the Bible says about Him. North Wind( link is to her hub) describes God's wrath under "God and Wrath." Voice CIW( link is to his hub) pulls no punches concerning God's wrath.
FIXING BIG GAPS IN YOUR CAPSULES:
For the first few days that the hub, Transforming the Fibonacci Sequence, was published I did not have the comments capsule in ; that has been fixed. A way to fix huge gaps in your capsules is given in the comments section of that hub.
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