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          E L-LOVELY-H math2-1     IInd  21-10-11     IIIrd  24-1-12     IVth  21-4-12     Vth  20-8-12     VIth 10-9-12






                                                                                  fJekJh^2 L ;hwk ns/ brkskosk

                2H1 cbB dh ;hwk (Limit of a Function)                                             B'N

                wzB fbU y = f (x) e'Jh cbB j? ns/ h 1 , h 2 , ........ h n , ........ fJZe XBkswe ;zfynktK dk ;w[Zu (a
                set of positive numbers) j? fi;dk wkB brksko xZN (continually decreasing) fojk j? Gkt
                                                h 1  > h 2  > h 3  > ....... > h n  > ....... > 0                                                        ...(1)
                ns/ fi;B{z,  n  B{z T[fus tZvk b? e/, fizBk ukjhJ/ T[Bk fJZSk nB[;ko S'Nk p/Dk ;ed/ jK. fJ;
                ;fEsh ftZu fit/A^fit/A h n  S'Nk j[zdk j? cbB d/ wkB,







                xNd/ iKd/ jB.
                i/eo fJZe ;zfynk A d/ tZb gqtfos (tends to) j[zd/ jB sK fJ; ;zfynk A B{z cbB f (x) dk x = a
                T[Zs/ dZyD gZy ;hwk (right hand limit) efjzd/ jB iK fJ; ;zfynk A B{z cbB f (x) dh, id'A x, a
                d/ tZb gqtfos j[zdk j?, dZyD gZy ;hwsk efjzd/ jB, ns/ fJ;B{z
                                   lim f (x) = A = f (a + 0)
                               x→+ 0
                                 a
                d/ d[nkok fbyd/ jB.
                fJZE/ n;hA x d/ e/tb T[jBK wkBK (values) T[Zs/
                jh ftuko ehsk j? fijV/ fe a s'A tZX (greater)
                jB. (fuZso ftZu e/tb a d/ ;Zi/ gk;/).

                j[D  n;hA  x  d/ T[jBK wkBK T[Zs/ ftuko eoKr/
                fijV/ a s'A xZN (smaller) jB (Gkt fuZso ftZu
                a d/ yZp/ gk;/).
                fit/A^fit/A h n  S'Nk j[zdk iKdk j? cbB
                                                    f (a – h 1 ), f (a – h 2 )......,f (a – h n ),......
                d/ wkB fJZe ;zfynk B d/ tZb gqtfos j[zd/ jB. fJ; ;zfynk B B{z cbB f (x) dh x = a T[Zs/ tkw
                gZy ;hwk (left hand limit) efjzd/ jB ns/ fJ;B{z
                 lim  f (x) = B = f (a ^ 0)
                 x→− 0
                   a
                d/ d[nkok fbyd/ jB.
                i/eo A = B Gkt
                               lim  f (x) =  lim  f (x)
                            x→+ 0      x→− 0
                                         a
                              a
                sK A B{z f (x) dh x = a T[Zs/ ;hwk efjzd/ jB.
                ;w[Zu h 1 , h 2 ,........, h n ,........ fJZe eqw (sequence) j? fi;dh ;hwk 0 j?. fJ;h soQK d{ik eqw (2)
                pDkT[Adk j?. fJZE/ ftP/p fXnkB d/D :'r rZb fJj j? fe ;hwk gqkgs j'D (limit to exist) d/ bJh f
                (a + h n ) eqw (1) dh soQK d/ jo soQK d/ eqw d/ bJh, ;zfynk A B{z gqtfos j'Dk ukjhdk j?. Gkt
                f(a - h n ) – A dk nzeVk nzso, h n  B{z T[fus S'Nk u[De/, fizBk ukjhJ/ fJZSk nB[;ko T[Bk S'Nk eo
                ;ed/ jK. a + h n  {iK a – h n } = x ns/  xa−  = h n  oZyD s/ n;hA ;hwk B{z j/m nB[;ko gfoGkfPs
                eo ;ed/ jK^


                                            LOVELY PROFESSIONAL UNIVERSITY                                               35
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