Section The Precise Definition Of A Limit

Transcription

1 Section The Precise Definition Of A imit Introduction So fr we hve tken n intuitive pproch to the concept of limit. In this section we will stte the forml definition nd use this definition to prove limit eists. Proofs A mthemticl proof is " convincing rgument epressed in the lnguge of mthemtics." (From How to Red nd Do Proofs b Dniel Solow.) A proof consists of mthemticl sttements tht re connected b rules of logic to produce the desired result. An emple of rule of logic is tht the sttement "A nd B" is true if both the sttements A nd B re true. Another rule of logic is tht the sttement "if A, then B" is flse onl when the sttement A is true nd the sttement B is flse. You will see most theorems stted in the form "if A, then B." Emple: Prove tht if n is n even integer, then n 2 is n even integer. The Concept of the Rigorous Definition of imit Recll how defined limit erlier. We s tht is the limit of f H s pproches some number, written lim Ø f H = if we cn mke the vlues of f H s close to s we like b tking vlues of close to.

2 2 ecture_02_04.nb Geometricll, we cn visulize this informl definition s follows. Consider the following grph. We s tht is the limit of f H s pproches some number if we cn mke the vlues of f H s close to s we like b tking vlues of close to. If "we cn mke the vlues of f H close to " then this mens tht if we tke the numbers c nd d close to the number such tht c d then we hve c f H d which mens tht prt of our grph of = f H is between the horizontl lines = c nd = d. The prt of the grph tht is between the horizontl lines = c nd = d is tht prt we obtin b "tking vlues of close to. " If the vlues re close to mens tht we cn tke numbers m nd n so tht m n nd m n. Thus we cn mke f H fll within c nd d if we tke between m nd n. This leds us to our forml definition. The Rigorous Definition of imit Definition 1 et f be function defined on some open intervl tht contins the number, ecept possibl t itself. Then we s tht the limit of f H s pproches is, nd we write lim Ø f H = if for ever number e >0 there eists number d >0 such tht if 0 < - <d then f H - <e

3 ecture_02_04.nb 3 Emple: Consider the limit lim Ø1 H2 + 1 = 3.. Find n open intervl bout = 1 so tht f H - <e, where f H = nd e=0.75. Repet with e=0.25 b. Wht vlue should be chosen for d so tht if 0 < - <dthen f H - <e where f H = 2 + 1, = 1, nd e=0.75? Repet with e=0.25. c. Prove tht lim Ø1 H2 + 1 = 3.

4 4 ecture_02_04.nb Emple: Prove tht lim Ø3 H3-7 = 2. Emple: Prove tht lim Ø2 2 = 4. Emple: Prove tht lim Ø1 1 = 1.

5 ecture_02_04.nb 5 Definition 3 - eft-hnd imit lim Ø - f H = if for ever number e >0 there eists number d >0 such tht if -d< < then f H - <e Definition 4 - Right-Hnd imit lim Ø - f H = if for ever number e >0 there eists number d >0 such tht if < < +d then f H - <e Infinite imits We cn define infinite limits rigorousl. Definition 5 et f be function defined on some open intervl tht contins the number, ecept possibl t itself. Then lim Ø f H = mens tht for ever positive number M there is positive number d such tht if 0 < - <d then f H > M Definition 6 et f be function defined on some open intervl tht contins the number, ecept possibl t itself. Then lim Ø f H =- mens tht for ever negtive number M there is positive number d such tht if 0 < - <d then f H < M imits t Infinit Definition 7 et f be function defined on some intervl H,. Then lim Ø f H = mens tht for ever number e>0 there is corresponding number N such tht if > N then f H - <e Definition 8 et f be function defined on some intervl H-,. Then lim Ø- f H = mens tht for ever number e>0 there is corresponding number N such tht if < N then f H - <e ü Definition 9 et f be function defined on some intervl H,. Then lim Ø f H = mens tht for ever positive number M there is corresponding positive number N such tht if > N then f H > M