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VED1
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

noEPk;soh dk rfDs

B'N gfoGkPk L x = a T[Zs/ cbB f (x) dh ;hwk e'Jh ;zfynk (wzB fbU A j[zdh j? fi; ftZu fJj
r[D j[zdk j? fe x d/ T[jBK ;ko/ wkBK d/ bJh fiBQK d/ Gkt x - a dk ;zfynkswe
wkB) T[fus o{g s'A S'Nh (gozs{ Iho' BjhA) j[zdh j?, Gkt f (x) – A dk
;zfynkswe wkB fJZSk nB[;ko S'Nk j[zdk j?.
;hwk B{z j/mbh gfoGkPk d[nkok th gofGkfPs eo ;ed/ jK^
;hwk dh d{ih gfoGkPk

id'A (id'A x, a d/ tZb gqtfos j[zdk j?) cbB f (x) dh ;hwk e'Jh ;zfynk (wzB fbU
A) j[zdh j? fi; ftZu fJj r[D j[zdk j? fe fe;h ;t/S XBkswe S'Nh s'A S'Nh ;zfynk ε d/
bJh fJZe d{ih Iho' Bkb tZvh ;zfynk δ gqkgs ehsh ik ;e/ fi;d/ bJh < ε, x
d/ T[jBK ;ko/ wkBK d/ bJh

H

B'N; i/eo x = a T[Zs/ f (x) dh ;hwk (limit) L j't/ sK fJ;B{z j/m nB[;ko fbfynk iKdk j?^
iK

2H2 dZyD gZy ns/ tkw gZy ;hwk (Right Hand and Left Hand Limits)

2H2H1 dZyD gZy ;hwk (Right hand limit)

id'A cbB dh ;hwk ;[szso uo d/ wkB d/ ;Zi/ gk;/
s'A gqkgs ehsh iKdh j? sK T[;B{z cbB dh dZyD
gZy ;hwk (R. H. L.) efjzd/ jB ns/ ;Zi/ gk;/ d/
bJh XzB fuzBQ dk gq:'r eod/ j'J/ ;ze/se o{g ftZu
j/m nB[;ko fbyd/ jB^

dZyD gZy ;hwk & f (a + 0) = f (x) = l 1 .

2H2H2 tkw gZy ;hwk (Left hand limit)
id'A cbB dh ;hwk ;[szso uo d/ wkB d/ yZp/ gk;/ s'A gqkgs ehsh iKdh j? sK T[;B{z tkw gZy
;hwk (L.H.L.) efjzd/ jB ns/ yZp/ gk;/ d/ bJh (^) fuzBQ dk gq:'r eod/ j'J/ ;ze/se o{g ftZu
j/m nB[;k fbyd/ jK^

tkw gZy ;hwk & f (a – 0) = f (x) = l 2 .

2H3 cbB dh ;hwk dZyD gZy ns/ tkw gZy ;hwk gsk eoB dh fefonk
ftXh
(Working Rules for Finding Right Hand Limit and Left Hand
Limit)

(i) dZyD gZy ns/ tkw gZy ;hwk gsk eoB d/ bJh cbB ftZu uo x d/ ;EkB s/ eqw nB[;ko
(x + h) ns/ (x - h) ;Ekfgs eo'.
(ii) fJ; soQK (i) s'A gqkgs cbB x B{z fdZs/ j'J/ fpzd{ (wzB fbU a) Bkb ;Ekfgs eo'.

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