Limits The notion of a limit is fundamental to the stud of calculus. It is one of the primar concepts that distinguishes calculus from mathematical subjects that ou saw prior to calculus, such as algebra and geometr. We use limits to discuss (slides 1-3) the derivative as a limiting slope the integral as a limiting sum of areas an infinite sum as a limit of finite partial sums What is the limit of a function? Intuitivel, we want the limit to sa that as gets closer to some value a, 0< a <δ the function values get closer to L. L < To rigorousl define close, we need a distance measure, which is, absolute value. Definition The limit of as approaches a is equal to L if for ever >0, there eists a δ>0suchthat In this case, we write Remarks 0 < a <δimplies L <. lim =L. In general, the value of δ will depend on the value of. That is, we will alwas begin with a value of and then determine an appropriate corresponding value for δ. (slide 4) Math 30G - Prof. Kindred - Lecture 2 Page 1
Note that in the def n of limit we have 0 < a <δ and not 0 a <δ. The behavior of atthepoint = a has no impact on the eistence of the limit of as approaches a. Eamples L The function could be undefined at = a. a L The function could be a value other than L at = a (so f(a) = L). a In both of the above cases, the limit as a eists and lim =L. Let s tr an eample to better understand the interpla between and δ. Math 30G - Prof. Kindred - Lecture 2 Page 2
Eample =2 and we are interested in lim 3 6 (slide 5) δ-band 3 -band If we believe lim 3 =6,then according to limit def n, given an >0, we should be able to find a δ>0suchthat for all -values in the δ-band (ecluding =3) = the corresponding function values lie inside the -band Sa =0.1 take suggestion from student If we make δ =.05, then we have 0 < 3 <.05 -values in δ-band = L = 2 6 =2 3 < 2(.05) =.1 -values in -band generalization = for a given >0, choose δ = 2 value of δ depends on Proof (that lim 3 =6): Suppose>0isgiven.Letδ = 2. Then, if 0 < 3 <δ,itfollowsthat 6 = 2 6 =2 3 < 2 2 =. Therefore, b the definition of limit, lim 3 =6. Math 30G - Prof. Kindred - Lecture 2 Page 3
Eample Prove that lim 1 (5 3) = 2. Scratch work: work backwards to find δ given > 0, we want (5 3) 2 < we get to bound 1, so be on the lookout for this epression we have 5 5 < 5( 1) < 5 1 < 1 < 5 Proof: Given >0, let δ = 5. Then, if 0 < 1 < 5,wehave that (5 3) 2 = 5 5 =5 1 < 5 5 =. Thus, it must be that lim(5 3) = 2. 1 How might a limit of a function fail to eist? M L If limit were L, M, orsome other value, convince ourself that there eists an >0such that no valid δ would eist. Math 30G - Prof. Kindred - Lecture 2 Page 4
However, we can define one-sided limits: left-hand limit right-hand limit lim =L lim =M + Of course, there are other reasons wh a limit ma fail to eist: (slide 6) oscillating behavior unbounded behavior To aid us in our proofs for limits, we have some powerful tools. Theorem. Suppose that lim =L and lim g() =M. Then lim [ ± g()] = L + M, lim [g()] = LM, if M = 0, lim g() = L M. Furthermore, if g() for all, then L M. Aketoolintheproofoftheabovetheoremisthetriangleinequalit: a + b a + b. Math 30G - Prof. Kindred - Lecture 2 Page 5
If time remains... What about limits for multivariable functions? Eamples Consider limit of f(, ) = 2 + 3 when (, ) (0, 0). (slide 7) (distance is measured b instead of ) Picture for a function f : R 2 R 2 If (, ) stasinsidethe δ-ball, then f(, ) shouldstainsidethe -ball. domain f codomain δ (a 1,a 2 ) ( 1, 2 ) Ke difference = There are infinitel man directions from which we can approach (a 1,a 2 ), as contrasted with onl two directions (left or right) for functions f : R R. and in all directions limits must eist and be equal Math 30G - Prof. Kindred - Lecture 2 Page 6
Slope of tangent line as a limit of slopes of secants Limits Math 30G, Calculus Professor Kindred September 7, 2012 Integral as limiting sum of areas Infinite sum as a limit of finite partial sums sequence of partial sums definition of infinite sum Determining delta from epsilon Limit of = 2 as approaches 3 δ-band 6 -band 3
Other reasons wh a limit ma not eist Multivariable limits 1.0 0.5 0.10 0.05 0.05 0.10 0.5 1.0 f () = sin 1 oscillating behavior a cubic of parabolas f : R 2 R defined as f (,)= 2 + 3 f () = 1 2 unbounded behavior