Mgr. ubomíra Tomková. Limit
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- Marvin Sutton
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1 Limit I mathematics, the cocept of a "limit" is used to describe the behaviour of a fuctio as its argumet either gets "close" to some poit, or as it becomes arbitrarily large; or the behaviour of a sequece's elemets, as their idex icreases idefiitely. Limits are used i calculus ad other braches of mathematical aalysis to defie derivatives ad cotiuity. A GRAPHICAL EXAMPLE: Now, let's look at the graph of f(x)= x 1 ad see what happes! The x-axis is a horizotal asymptote... Let's look at the blue arrow first. As x gets really, really big, the graph gets closer ad closer to the x-axis which has a height of. So, as x approaches ifiity, f(x) is approachig. This is called a limit at ifiity. Now let's look at the gree arrow... What is happeig to the graph as x gets really, really small? The graph is agai gettig closer ad closer to the x-axis (which is.) It's just comig i from below this time. But what happes as x approaches? 1
2 Sice differet thigs happe, we eed to look at two separate cases: what happes as x approaches from the left ad what happes as x approaches from the right: ad Sice the limit from the left does ot equal the limit from the right... A GEOMETRIC EXAMPLE: Let's look at a polygo iscribed i a circle... If we icrease the umber of sides of the polygo, what ca you say about the polygo with respect to the circle? As the umber of sides of the polygo icreases, the polygo is gettig closer ad closer to the circle!
3 If we refer to the polygo as a -go, where is the umber of sides, we ca make some equivalet mathematical statemets. As gets larger, the -go gets closer to beig the circle. As approaches ifiity, the -go approaches the circle. The limit of the -go, as goes to ifiity, is the circle! The -go ever really gets to be the circle, but it will get pretty close! So close, i fact, that, for all practical purposes, it may as well be the circle. That's what limits are all about! Archimedes used this idea to fid the area of a circle before they had a value for PI! (They kew PI was the circumferece divided by the diameter... But, they did't have calculators back the.) Limit of a sequece Cosider the followig sequece: 1.79, 1.799, ,... We could observe that the umbers are "approachig" 1.8, the limit of the sequece. Formally, suppose x 1, x,... is a sequece of real umbers. We say that the real umber L is the limit of this sequece ad we write if ad oly if for every real umber > there exists a atural umber (which will deped o ) such that for all > we have x L <. Ituitively, this meas that evetually all elemets of the sequece get as close as we wat to the limit, sice the absolute value x L ca be iterpreted as the "distace" betwee x ad L. Not every sequece has a limit; if it does, we call it coverget, otherwise diverget. Oe ca show that a coverget sequece has oly oe limit. 3
4 SOME NUMERICAL EXAMPLES: EXAMPLE 1: Let's look at the sequece whose th term is give by. Recall, that we let = 1 to +1 get the first term of the sequece, we let = to get the secod term of the sequece ad so o. What will this sequece look like? 1/, /3, 3/4, 4/5, 5/6,... 1/11,... 99/1, /1,... What's happeig to the terms of this sequece? Ca you thik of a umber that these terms are gettig closer ad closer to? The terms are gettig closer to 1! But, will they ever get to 1? No! So, we ca say that these terms are approachig 1. It souds like a limit! The limit is 1. As gets bigger ad bigger, gets closer ad closer to EXAMPLE : Now, let's look at the sequece whose th 1 term is give by. What will this sequece look like? 1/1, 1/, 1/3, 1/4, 1/5,... 1/1,... 1/1,... 1/1,... As gets bigger, what are these terms approachig? That's right! They are approachig. How ca we write this i Calculus laguage? Examples: 1. ( 4 ) ( 5 )( ) lim = lim x lim = lim = = = lim x = 1 4 lim x =
5 Limit of a fuctio Suppose ƒ(x) is a real fuctio ad c is a real umber. The expressio: meas that ƒ(x) ca be made to be as close to L as desired by makig x sufficietly close to c. I that case, we say that "the limit of ƒ of x, as x approaches c, is L". Note that this statemet ca be true eve if. Ideed, the fuctio ƒ(x) eed ot eve be defied at c. Two examples help illustrate this. Cosider its limit of.4: f(1.9) f(1.99) f(1.999) f() as x approaches. I this case, f(x) is defied at ad equals f(.1) f(.1) f(.1) As x approaches, ƒ(x) approaches.4 ad hece we have. I the case where always the case., ƒ is said to be cotiuous at x = c. But it is ot Cosider the case where ƒ(x) is udefied at x = c. I this case, as x approaches 1, f(x) is udefied at x = 1 but the limit equals : f(.9) f(.99) f(.999) f(1.) f(1.1) f(1.1) f(1.1) udef Thus, x ca get as close to 1, so log as it is ot equal to 1, so that the limit of f(x) is. FORMAL DEFINITION A limit is formally defied as follows: Let f(x) be a fuctio defied o a ope iterval cotaiig c (except possibly at c) ad let L be a real umber. 5
6 meas that for each real there exists a real such that for all where,. I words, if ad oly if by takig x close eough to c we ca get f(x) arbitrarily close to L. The formal defiitio of a limit is sometimes called the delta-epsilo form because it uses the Greek letters delta () ad epsilo (). I practice, ad are just variables ad could be replaced by ay other letters. Properties of the Limit Each of the followig properties is prove usig the rigorous defiitio of the limit. Let lim stad for lim, lim, or + x c x c lim x c Assume lim f(x) ad lim g(x) both exist. (Uiqueess) If lim f(x) = L 1 ad lim f(x) = L, the L 1 = L. (Additio) lim [f(x) + g(x)] = lim f(x) + lim g(x) (Scalar multiplicatio) lim [c. f(x)] = c. lim f(x). (Multiplicatio) lim [f(x). g(x)] = = lim f(x). lim g(x). f ( x) lim f ( x) (Divisio) lim =, provided lim g(x). g( x) lim g( x) (Powers) lim [f(x)] = [lim f(x)] for ay positive iteger. I practice, much of the time we ca ``reaso out'' the value of a limit without explicitly usig the - defiitio. LIMITS OF FUNCTIONS AS X APPROACHES A CONSTANT The followig problems require the use of the algebraic computatio of limits of fuctios as x approaches a costat. Most problems are average. A few are somewhat challegig. All of the solutios are give WITHOUT the use of L'Hopital's Rule. If you are goig to try these problems before lookig at the solutios, you ca avoid commo 6
7 mistakes by givig careful cosideratio to the form durig the computatios of these limits. Iitially, may studets INCORRECTLY coclude that is equal to 1 or, or that the limit does ot exist or is or -. I fact, the form is a example of a idetermiate form. This simply meas that you have ot yet determied a aswer. Usually, this idetermiate form ca be circumveted by usig algebraic maipulatio. Such tools as algebraic simplificatio, factorisig, ad cojugates ca easily be used to circumvet the form so that the limit ca be calculated. As a result of these theorems, we see that for may fuctios f, A fuctio which has this property is called cotiuous. Polyomial fuctios ad ratioal fuctios are cotiuous. The followig examples demostrate how we ca evaluate limits of fuctios which are ot cotiuous by usig the above-metioed list of limit theorems. Examples: 1... (Circumvet the idetermiate form by factorig both the umerator ad deomiator.) 7
8 (Divide out the factors x -, the factors which are causig the idetermiate form. Now the limit ca be computed. ) 3. (Elimiate the square root term by multiplyig by the cojugate of the umerator over itself. Recall that (a b) (a + b) = a b (Divide out the factors x - 4, the factors which are causig the idetermiate form. Now the limit ca be computed. ). 4. (If you wrote that si (5x) = 5 si x, you are icorrect. Istead, multiply ad divide by 5, sice we wat to apply the well-kow fact that lim x si x = 1. x. 8
9 Limit of a fuctio at ifiity A related cocept to limits as x approaches some fiite umber is the limit as x approaches positive or egative ifiity. This does ot literally mea that the differece betwee x ad ifiity becomes small, sice ifiity is ot a real umber; rather, it meas that x either grows without boud positively (positive ifiity) or grows without boud egatively (egative ifiity). For example, cosider. f(1) = 1.98 f(1) = f(1) = As x becomes extremely large, the value of f(x) approaches, ad the value of f(x) ca be made as close to as oe could wish just by pickig x sufficietly large. I this case, we say that the limit of f(x) as x approaches ifiity is. I mathematical otatio, Formally, we have the defiitio if ad oly if for each > there exists a such that f(x) c < wheever x >. Note that the i the defiitio will geerally deped o. A similar defiitio applies for. If oe cosiders the domai of to be the exteded real umber lie, the the limit of a fuctio at ifiity ca be cosidered as a special case of limit of a fuctio at a poit. 9
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