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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