Tangent Lines. Limits 1

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1 Limits Tangent Lines The concept of the tangent line to a circle dates back at least to the earl das of Greek geometr, that is, at least 5 ears. The tangent line to a circle with centre O at a point A is simpl constucted b taking the perpendicular to the line through O and A at A. A O The conic sections parabolas, ellipses, and hperbolas, were discovered and studied about ears ago. The idea of a tangent line to such a curve was simple enough, since these curves can all be viewed as being shadows of circles, but the geometric construction of these tangent lines with the tools of the da was ver difficult. It was onl with the discover, or invention, of Calculus in the 7 th centur that the proper tools became available, and it then became possible to define the concept of tangent lines at points on a ver large class of curves.

2 Limits Velocit Again, the histor of the concept of velocit begins with the Greeks, especiall with Zeno s four paradoes of motion. See the Internet Encclopedia of Philosoph. Parado: Suppose an archer shoots an arrow with the velocit of metres/second at a fleeing enem metres awa who is running awa at the velocit of metres/second. As the arrow traverses the first metres, the enem travels metres. As the arrow traverses those metres, the enem moves one metre, and so on. Therefore the arrow never hits the enem. While this model of realit is not supported b eperience don t tr it!!! it serves to point out the problems underling the concept of velocit. The questions raised b Zeno and others were not successfull resolved until after not onl Calculus as a tool had been constructed, but the careful definition of the real numbers had been completed about The underling concept needed to understand the correct definitions of tangent lines and velocities is that of the limit of a function. We begin b tring to develop a visual feeling for the associated concepts.

3 Limits The Stud of Limits To man, this treatment of limits will appear to be in the wrong order. What we are going to attempt is to develop an intuitive geometric feeling for what limits are supposed to be b eamining the graphs of a few functions that are fairl eas to graph, especiall with the aid of a computer. However, the students should keep in mind that the reall interesting situations are those that arise when functions are complicated and mischievousl cause computers to misbehave. It is then that the mathematical techniques of limit calculation come to the rescue, and enable us to correctl sketch the graph of the function. A note on computer graphing: usuall we are luck when a computer has trouble graphing a function because we see a mess of vertical lines close to the trouble spot, or we get error messages about divisions b zero or square roots of negative numbers. When we are unluck, an incorrect but seemingl normal curve appears. It is then that the habit of scrupulous mathematical thinking pas off. So, our program is this: () Intuitive Ideas of Limits(we re in it alread) () Computational Techniques for finding limits(net lesson) () Rigorous Treatment of Limits(if ou start now, and keep at it, ou will finall feel ou understand this in about two or three ears!) There is a stor of Abe Lincoln as a oung man practising law in rural Illinois in the 85 s, reading and rereading Euclid s Elements as he rode on horseback from one town s courthouse to the net, until he felt that he completel understood all of it. The mental discipline was then invaluable to him in later ears. Thus, the stud of the rigorous treatment of limits can be epected to eercise a student s mind in the most beneficial wa.

4 4 Limits Consider the function f() =. Its domain is the set (, ) (, ). Since = +, the graph of = f() is the straight line with slope and -intercept, with the point (,) removed From the picture we can see that the closer gets to, the closer the value of f() gets to. In a situation like this, we sa the the limit of f() as approaches is, and in general: we sa that the limit as approaches a is L if the values of f()get arbitraril close to L as long as is close enough to a. We write lim f() = L a

5 Limits 5 Fact: If f() is a polnomial function, a rational function, an eponential function or a logarithmic function and a is in the domain of f, then lim a f() = f (a). Thus limits aren t ver interesting unless something special is going on with the function, such as a being outside the domain( as in our first eample), or a being at the boundar of the domain, or a being a point of transition of the function s definition from one formula to another in the case of a function defined with a multiline formula. { Eample: if < Let f() = if Here a = is in the domain, but the limit lim f() does not eist. a This function is known as the Heaviside function - - -

6 6 Limits We also have the concept of One-Sided Limits: One-Sided Limits we sa that the limit as approaches a is close enough to, and to the left of a. from the left is L if the values of f() get arbitraril close to L as long as We write lim f() = L. a we sa that the limit as approaches a from the right is L if the values of f() get arbitraril close to L as long as is close enough to, and to the right of a. We write lim f() = L. a + Sometimes we will refer to the ordinar limit lim a f() as a two-sided limit. Fact: If lim f() eists then both lim f() = L and lim f() = L eist. Furthermore, the are both equal to a a a + lim f(). Conversel, if both one-sided limits eist and are equal, then so does the two-sided limit, and it equals the a common value of the two-one-sided lmits.

7 Limits 7 Eample: In the case of the Heaviside function, we have f() = ifa<, lim a lim a lim f() = ifa>, f() =, lim f() =, + so the two-sided limit eists everwhere ecept at a = Eample: Let f() = { if if = Here a = is in the domain, and the limit lim a f() =, but f() =, which is not the value of the limit

8 8 Limits Infinite Limits If the value of a function tends to + or as approaches a number a, we will sa that the limit is before we will have one- and two-sided limits. infinite. As Eample: Suppose f() =. Then if a, we have lim f() =, but if a =, the (two-sided) limit does not eist. a a However, we have lim f() =, and lim f() =+, + and we can also talk about the limits at infinit : lim f() = lim f() =. Note the attachment of arrows to the function s graph to indicate such infinite limits

9 Limits 9 These infinite limits lead us to define Asmptotes asmptotes of a function: If, for a finite number a, lim a f() =± or lim f() =±, we sa that the line = a is a vertical asmptote of f. a + If lim f() = h or lim f() = k for finite numbers h or k, we sa that the lines = h or = k are of f. In the figure, the function sketched is f() = +. horizontal asmptotes

10 Limits There is another tpe of asmptote (not usuall covered in introductor Calculus) called a are numbers m and b such that lim f() (m + b) =, ± then the line = m + b is a slant asmptote. slant asmptote: if there In the figure, the function sketched is f() = ( ) +. The equation of the slant asmptote is =

11 Limits More Eamples Eample: Let f() = sin π if <. It s graph is show below: The limit is undefined at =. f(( t,)) = [, ]. As a matter of fact, for an number t (, ) we have f((,t)) = [, ], and

12 Limits Eample: Let f() = sin π if <. It s graph is show below: The limit is at =.