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'` `ecg
ilaib aia` :lbxzn
2008 lixt`a 9
deey dcina zetivx 1lkl m` D a deey dcina dtivx f(x) ik xn`p .D megza zxcbend divwpet f(x) idz
:y jk δ > 0 miiw ε > 0
∀x1, x2 ∈ D : |x1 − x2| < δ ⇒ |f(x1)− f(x2)| < ε
:1 libxz
.[1,∞) rhwa y"na dtivx f(x) =√
x ik egiked
:oexzt
if` |x1 − x2| < δ miniwnd x1, x2 ∈ [1,∞) lkly jk δ > 0 `evnl jixv ε > 0 idi:okle |f(x1)− f(x2)| < ε miiwzn
|f(x1)−f(x2)| = |√
x1−√
x2| =∣∣∣∣(√x1 −
√x2)(
√x1 +
√x2)
(√
x1 +√
x2)
∣∣∣∣ =
∣∣∣∣ x1 − x2√x1 +
√x2
∣∣∣∣ < |x1−x2|
.|f(x1)− f(x2)| < |x1 − x2| < δ miiwzi f` δ = ε gwip m` jkl i`
:uiytil i`pz
lkly jk K iynn reaw miiw ik gippe (a, b) rhwa zxcbend divwpet f(x) idz:miiwzn (a, a) rhwa f(x) zecewp izy
|f(x1)− f(x2)| ≤ K|x1 − x2|
:dprh
dtivx f(x) if` megza zecewp izy lkl uiytil i`pz zniiwn f(x) y dxwna.D megza y"na
1
ilaib aia`'` `"ecg
:dgked
jk K ∈ R miiw if` uiytil i`pz zniiwne D megza zxcbend divwpet f(x) idz:miiwzn x1, x2 ∈ D lkly
|f(x1)− f(x2)| ≤ K|x1 − x2|
:okle |x1 − x2| < δ miiwzn x1, x2 ∈ D lkl ik lawp f`e δ = εK
xgape ε > 0 idi
|f(x1)− f(x2)| ≤︸︷︷︸Lipchitz
K|x1 − x2| < K · ε
K= ε
.D megza y"na dtivx f(x) jkl i`
:htyn
.D megza dtivx `id if` D megza y"na dtivx f(x) m`
:xehpw htyn
.y"na ea dtivx [a, b] xebq rhwa dtivx divwpet
:dprh
{xn}∞n=1, {yn}∞n=1 ∈ D zexcq izy lkl if` D rhwa y"na dtivx f(x) m`.limn→∞(f(xn)− f(yn)) = 0 miiwzn if` limn→∞(xn − yn) = 0 zeniwnd
:dgked
:miiwzn x, y ∈ D lkly jk δ > 0 miiw okle D a y"na dtivx f(x) .ε > 0 idi
|x− y| < δ ⇒ |f(x)− f(y)| < ε
|xn − yn| < δ miiwzn n > N lkly jk N miiw if` limn→∞(xn − yn) = 0 e zeidokle
.limn→∞(f(xn)− f(yn)) = 0 xnelk |f(xn)− f(yn)| < ε miiwzi n > N lkl
:milibxz
:y"na dtivx `id m`d eraw ze`ad zeivwpetdn zg` lkl
(−∞,∞) rhwa sin x2 .1
(0, 1) rhwa sin πx
.2
2
ilaib aia`'` `"ecg
(−∞,∞) rhwa 2 sin x− cos x .3
[0,∞) rhwa x sin x .4
:zeaeyz
1. `l , 2. `l , 3. ok , 4. `l
zxfbp 1.1
xy`k) x = x0 +∆x onqp .x0 dcewpd ly zniieqn daiaqa zxcbend divwpet f(x) idzm` f`e ∆y = f(x0 + ∆x) − f(x0) mb onqp .(x0 dcewpl dpzyn zixtqn ztqez ∆x
:`ad leabd miiw
lim∆x→0
∆y
∆x= lim
∆x→0
f(x0 + ∆x)− f(x0)
∆x= L
mpyi f ′(x) = L onqpe x0 a f(x) ly zxfbpd epid leabd .x0 a dxifb f(x) ik xn`p if`.y′, dy
dx, df
dxoebk mitqep mipeniq mb
:dxrd
wiyn xyi `xwp f ′(x0) l deey eretiy xy` (x0, f(x0)) dcewpd jxc xaerd xyidzecewpd z` xagnd ewd xveiy zeiefd α idz .∆y
∆x= tan θ miiwzn xnelk
:ik lawp ileabd avna f`e (x0, f(x0)) e (x0 + ∆x, f(x0 + ∆x))
lim∆x→0
∆y
∆x= lim
∆x→0tan θ = tan α
e y− y0 = ∆y zexg` milina e` y− y0 = f ′(x0)(x−x0) `id wiynd xyid z`eeyn. ∆y
∆x= f ′(x0) ileabd avna f`e x− x0 = ∆x
:htyn
.ef dcewpa dtivx `id if` x0 dcewpa dxifb f(x) m`
:dxrd
.da dxifb dcewpa dtivxy divwpet lk `le oekp eppi` jtdd
:`nbec
z` wecap ik x0 = 0 a dxifb dppi` j` xyid lk lr dtivx f(x) = |x| divwpetd
3
ilaib aia`'` `"ecg
:ik lawpe leabd
lim∆x→0
f(0 + ∆x)− f(0)
∆x= lim
∆x→0
|∆x| − 0
∆x= lim
∆x→0
|∆x|∆x
:ik miiw `l leabd la`
lim∆x→0+
|∆x|∆x
= lim∆x→0+
∆x
∆x= 1
lim∆x→0−
|∆x|∆x
= lim∆x→0−
−∆x
∆x= −1
:dxcbd
g(x) = f ′(x) divwpetd if` (a, b) rhwa dcewp lka dxifbd divwpet f(x) idz.(a, b) a f(x) ly zxfbp z`xwp x ∈ (a, b) xear
:zeiccv cg zexfbpzniiw m` f(x) ly zipni zxfbp .x0 ly zipni daiaqa zxcbend divwpet f(x) idz
.f ′+(x0) onqpe lim∆x→0+f(x0+∆x)−f(x0)
∆xmiiw m` leabd dpid
.zil`ny zxfbp iabl dneca [∗]
:htyn
zeniiw m` wxe m` x0 a dxifb f(x) ,x0 ly daiaqa zxcbend divwpet f(x) idz.f ′−(x0) = f ′(x0) = f ′+(x0) xnelk x0 a zeiccv cgd zexfbpd zeeye
:zeidl zxcben x = a dcewpa y = f(x) divwpetd ly zipnid zxfbpd
f ′+(a) = limh→0+
f(a + h)− f(a)
h= lim
h→0+
f(x)− f(a)
x− a
:zeidl zxcben x = a dcewpa y = f(x) divwpetd ly zil`nyd zxfbpd
f ′−(a) = limh→0−
f(a + h)− f(a)
h= lim
h→0−
f(x)− f(a)
x− a
:1 libxz
:divwpetd dpezp{2x3 − 5x , x < 0
x + 2 , x ≥ 0
4
ilaib aia`'` `"ecg
.x = 0 dcewpa dxcbd it lr exfib
:oexzt
dcewpa dtivx `l divwpetde oeikny dcaera ynzydl `id dpey`x jxczeiccv cg zexfbp ly dxcbd ici lr `id dipy jxc .my dxifb `l mb `id f` x = 0
:ok`e
f ′+(0) = limh→0+
f(0 + h)− f(0)
h= lim
h→0+
(h + 2)− 2
h= lim
h→0+1 = 1
f ′−(0) = limh→0−
f(0 + h)− f(0)
h= lim
h→0−
(2h3 − 5h)− 2
h= lim
h→0+2h2 − 5− 2
h= ∞
.zniiw dppi` f ′(0) okle f ′+(0) 6= f ′−(0) jkl i`
:zeillk dxifb ze`gqep
:yxtde mekq z`gqep [1]
(f(x)± g(x))′ = f ′(x)± g′(x)
:ze`nbec
[`]f(x) = −5x3 + 7x2 + 2 ⇒ f ′(x) = −15x2 + 14x
[a]f(x) = 7x3 + 2x−4 ⇒ f ′(x) = 21x2 − 8x−5
:dltkn z`gqep [2]
(f(x) · g(x))′ = f ′(x) · g(x) + f(x) · g′(x)
:ze`nbec
[`]
f(x) = (3x2 − 5x)2 = (3x2 − 5x)(3x2 − 5x) ⇒ f ′(x) = (6x− 5)(3x2 − 5x)+
(3x2 − 5x)(6x− 5)
5
ilaib aia`'` `"ecg
[a]
f(x) = (2x + 7)(x2 − 3x) ⇒ f ′(x) = 2(x2 − 3x) + (2x− 3)(2x + 7)
:dpnd z`gqep [3](f(x)
g(x)
)′=
f ′(x) · g(x)− f(x) · g′(x)
g2(x)
:ze`nbec
[`]
f(x) =x2 + 2x
x− 5⇒ f ′(x) =
(2x + 2)(x− 5)− (x2 + 2x) · 1(x− 5)2
=x2 − 10x− 10
(x− 5)2
[a]
f(x) = −x4 − 3x
x2 − 1⇒ f ′(x) = −
((4x3 − 3)(x2 − 1)− (2x)(x4 − 3x)
(x2 − 1)2
)=
=−2x5 + 4x3 − 3x2 − 3
(x2 − 1)2
:zxyxyd llk - zakexn divwpet ly zxfbp
:f` f(x) = g(h(x)) xnelk zakexn divwpet `id f(x) divwpetd m`
f ′(x) = g′(h(x)) · h′(x)
.ziniptd z` f`e zipevigd divwpetd z` xefbl yi xnelk
:ze`nbec
[`]f(x) = (3x4 − 5x)7 ⇒ f ′(x) = 7(3x4 − 5x)6 · (12x3 − 5)
[a]
f(x) =x
(ax + b)4⇒ f ′(x) =
1 · (ax + b)4 − x · 4(ax + b)3 · a(ax + b)8
=
(ax + b)3(ax + b− 4xa)
(ax + b)8=
b− 3xa
(ax + b)5
6
ilaib aia`'` `"ecg
[b]
f(x) =
(x + 1
2x + 1
)4
⇒ f ′(x) = 4
(x + 1
2x− 1
)3
·(
1 · (2x + 1)− 2(x + 1)
(2x + 1)2
)=
= 4
(x + 1
2x− 1
)3
·(
−1
(2x + 1)2
)=−4(x + 1)3
(2x + 1)5
:miyxey ly zxfbp
.epi`xy dpey`xd `gqepd itl xefbp f`e m√
xn = xnm y dcaera ynzyp
:ze`nbec
[`]
f(x) =√
x = x12 ⇒ f ′(x) =
1
2· x−
12 =
1
2√
x
[a]
f(x) =
√5x2 − 2
√x = (5x2−2
√x)
12 ⇒ f ′(x) =
1
2·(5x2−2
√x)−
12 ·(10x−2· 1
2√
x) =
=10x− 1√
x
2√
5x2 − 2√
x
[b]
f(x) =
√x2 + x
2x + 1=
(x2 + x)12
2x + 1⇒ f ′(x) =
12· (x2 + x)−
12 · (2x + 1) · (2x + 1)− (x2 + x)
12 · 2
(2x− 1)2=
=12· (x2 + x)−
12 · (2x + 1)2 − 2(x2 + x)
12
=
1√x2 + x
− 2√
x2 + x
(2x + 1)2
:zeixhnepebixh zeivwpet ly zexfbp
(cos x)′ = − sin x [a] (sin x)′ = cos x [`]
(cot x)′ = − 1sin2 x
[c] (tan x)′ = 1cos2 x
[b]
:ze`nbec
[`]f(x) = cos 2x ⇒ f ′(x) = − sin 2x · 2 = −2 sin 2x
7
ilaib aia`'` `"ecg
[a]
f(x) = sin4 2x ⇒ f ′(x) = 4 sin3 2x · cos 2x · 2 = 8 sin3 2x cos 2x
[b]
f(x) = x2 tan2 4x ⇒ f ′(x) = 2x tan4 2x + x2 · 2 tan 4x · 1
cos2 4x· 4
[c]
f(x) = sin3 x cos x ⇒ f ′(x) = 3 sin2 x · cos x · cos x + sin3 x · (− sin x) =
= 3 sinx cos2 x− sin4 x
[d]
f(x) = cos(x + sin 3x) ⇒ f ′(x) = − sin(x + sin 3x) · (1 + cos 3x · 3)
[e]
f(x) = sin
(x + 1
x− 1
)⇒ f ′(x) = cos
(x + 1
x− 1
)· 1 · (x− 1)− 1(x + 1)
(x− 1)2
= cos
(x + 1
x− 1
)· −2
(x− 1)2
[f]
f(x) =2 tan x
1 + cot2 x=
2 sin xcos x1
sin2 x
= 2 sin x cos x = sin 2x ⇒ f ′(x) = 2 cos 2x
:zeinzixbel zeivwpet ly zexfbp
:`id zxfbpd illk ote`a
(loga(f(x)))′ =f ′(x)
f(x)· loga e
.(ln(f(x)))′ = f ′(x)f(x)
`ed ihxt dxwn
:ze`nbec
8
ilaib aia`'` `"ecg
[`]
f(x) = log5(x2 − 4) ⇒ f ′(x) =
2x
x2 − 4· log5 e
[a]
f(x) = ln
√x− 3
x− 2= ln
(x− 3
x− 2
) 12
=1
2ln
(x− 3
x− 2
)=
1
2(ln(x− 3)− ln(x− 2))
⇒ f ′(x) =1
2
(1
x− 3− 1
x− 2
)[b]
f(x) = ln(5x2 − 7x) ⇒ f ′(x) =10x− 7
5x2 − 7x
[c]
f(x) = ln3(sin x) ⇒ f ′(x) = 3 ln2(sin x) · cos x
sin x= 3 cot x ln2(sin x)
[d]
f(x) = sin2(ln x) ⇒ f ′(x) = 2 sin(ln x) · cos(ln x) · 1
x=
sin(2 ln x)
x
[e]
f(x) = (ln x)(ln ln x) ⇒ f ′(x) =1
xln ln x + ln x ·
1x
ln x=
ln ln x + 1
x
[f]
f(x) = ln2 sin2(x2) ⇒ f ′(x) = 2 ln sin2(x2) · 1
sin2 x2· (2 sin x2 cos x2 · (2x)) =
= 8x cos2 x2 ln sin2(x2)
:zeikixrn zeivwpet ly zexfbp
:`id zxfbpd illk ote`a(af(x)
)′= ln a ·
(af(x)
)· f ′(x)
9
ilaib aia`'` `"ecg
.(ef(x)′
= ef(x) · f ′(x) `ed ihxt dxwn
:ze`nbec
[`]f(x) = e2x ⇒ f ′(x) = (ln e) · e2x · 2 = 2e2x
[a]
f(x) = 53x2 ⇒ f ′(x) = (ln 5) ·(53x2
)· (6x) = 6x ln 5 · 53x2
[b]
f(x) = e√
x ⇒ f ′(x) =1
2√
x· e√
x
[c]
f(x) = esin2x ⇒ f ′(x) = esin2x · (2 sin x · cos x) = esin2x sin 2x
[d]
f(x) = ex ln x ⇒ f ′(x) = ex ln x + ex · 1
x= ex
(ln x +
1
x
)[e]
f(x) = ln(xex) ⇒ f ′(x) =1 · ex + xex
xex=
1 + x
x
:ik lawpe mitb`d ipy lr ln lirtp if` y = x−x idz [f]
ln y = ln x−x = −x ln x
:ik lawpe mitb`d ipya xefbp zrke
y′
y= − ln x− x · 1
x= − ln x− 1 ⇒ y′ = y(− ln x− 1) = x−x(− ln x− 1)
:ik lawpe mitb`d ipy lr ln lirtp if` y = xln x idz [g]
ln y = ln xln x = ln x ln x = ln2 x
:ik lawpe mitb`d ipya xefbp zrke
y′
y= 2 ln x · 1
x⇒ y′ = y
2 ln x
x=
2xln x ln x
x= 2xln x−1 ln x
10
ilaib aia`'` `"ecg
dketd divwpet ly zxfbp 1.1.1
:htyn
e x0 a dxifb f(x) m` .x0 dcewpd zaiaqa dtivxe dkitd divwpet y = f(x) idzy0 = f(x0) dcewpa dxifb x = g(y) dly dketdd divwpetd mb if` f ′(x0) 6= 0
.g′(y0) = 1f ′(x0)
miiwzne
:1 libxz
.(ax)′ z` eayg
:oexzt
dly dketdd divwpetde xyid lk lr dxifbe dkitd divwpet `idy y = ax onqp:ik lawp htynd itl jkl i` g′(y) = 1
y ln aokle g(y) = loga y = ln y
ln a`id
(ax)′ =1
g′(y)= y ln a = ax ln a
.(af(x))′ = 1g′(y)
= y ln a = af(x) · f ′(x) ln a illk ote`a
ddeab xcqn zexfbp 1.1.2
divwpet lawp (a, b) zecewp lka f(x) z` xefbp m`e (a, b) a dxifb divwpet f(x) idzm` .(a, b) a f(x) ly dpey`xd zxfbpd z`xwp f ′(x) .(a, b) a zxcben mby f ′(x) dycgf(x) ly diipyd zxfbpd efe (a, b) a zxcben mb f ′′(x) dzxfbp if` (a, b) a dxifb mb f ′(x)
.f (n) onqpe f(x) ly n xcqn zxfbp xicbdl ozip jke (a, b) a
:milibxz
:1 dl`y:ze`ad zeivwpetd z` exfb
f(x) = e−x2ln√
1 + x3 [a] f(x) = 3√
x [`]
f(x) = 3
√x2+11−x [c]
(0, π
2
)rhwa f(x) = xsinx + (sinx)x [b]
f(x) = arcsin(
x√1+x2
)[e] f(x) = 3
√x +
√x [d]
11
ilaib aia`'` `"ecg
f(x) = arcsin(cos x) [g] f(x) = 1√3x2+1
[f]
:2 dl`y
:ze`ad zeprhd z` ekixtd e` egiked
`l g′(a) e zniiw h′(a) day a dcewpa dxifb `l F (x) = g(x) + h(x) divwpetd [`].zniiw
g′(a) e zniiw `l h′(a) day a dcewpa dxifb `l F (x) = g(x) + h(x) divwpetd [a].zniiw `l
`l g′(a) e zniiw h′(a) day a dcewpa dxifb `l F (x) = g(x) · h(x) divwpetd [b].zniiw
g′(a) e zniiw `l h′(a) day a dcewpa dxifb `l F (x) = g(x) · h(x) divwpetd [c].zniiw `l
:3 dl`y
:ze`ad zeprhd z` ekixtd e` egiked
. limn→∞
n(f(x + 1
n
)− f(x)
)= f ′(x) miiwzn if` dxifb f(x) m` [`]
.dxifb f(x) if` limn→∞
n(f(x + 1
n
)− f(x)
)= L iteqd leabd miiw m` [a]
.dneqg f(x) da a ly daiaq zniiw m`d a dcewpa dxifb f(x) idz [b]
:4 dl`y
.a a dtivx ϕ(x) xy`k f(x) = (x− a)ϕ(x) y oezp m` f ′(a) z` e`vn [`]
.a a 0 n dpeye dtivx ϕ(x) e f(x) = |x− a|ϕ(x) xy`k zniiw f ′(a) m`d [a]
.a1, a2, · · · , an zecewpa dxifb `l xy` dtivx divwpet epa [b]
:5 dl`y
divwpetd s ly mikxr eli` xear ewca
f(x) =
xs sin 1x , x 6= 0
0 , x = 0
12
ilaib aia`'` `"ecg
.dzxfbp eayge 0 a dxifbe dtivx
:6 dl`y
.zeivwpet od u(x), v(x) xy`k u(x)v(x) ly zxfbpd z` e`vn
:zeaeyz:1 dl`y:'` sirq
:ik lawpe millkd itl xefbp
3√
x = x13 ⇒
(x
13
)′=
13x
13−1 =
13x−
23 =
13 3√
x2
:'a sirq
:ik lawpe zxyxyde dltknd llka ynzyp
f(x)′ =(e−x2
)′ln√
1 + x3 + e−x2(ln√
1 + x3)′
=(e−x2
· (−2x))
ln√
1 + x3+
e−x2(
1√1 + x3
· (√
1 + x3)′)
= −2xe−x2ln√
1 + x3 + e−x2· 1√
1 + x3· 12
(√1 + x3
)− 12 · 3x2
= −2xe−x2ln√
1 + x3 +32x2e−x2
· 11 + x3
:'b sirq
:`ad gezitd z` d`xp
(sinx)x = ex ln(sin x) , xsin x = eln xsin x
= esin x ln x
:ik lawpe millkd itl xefbp zrk
(xsin x)′ = (esin x ln x)′ = esin x ln x · (sinx lnx)′ = esin x ln x
(cos x lnx +
sinx
x
)
((sinx)sin x)′ = (ex ln(sin x))′ = ex ln(sin x) · (x ln(sin x))′ = ex ln(sin x)
(ln(sin x) + x · 1
sinx· cos x
)
(sinx)x(ln(sin x) +
x cos x
sinx
)
13
ilaib aia`'` `"ecg
:lawp lkd jqa okl
(xsin x + (sinx)sin x)′ = xsin x
(cos x lnx +
sinx
x
)+ (sinx)x
(ln(sin x) +
x cos x
sinx
)
:'c sirq
:ik lawpe millkd itl xefbp(3
√x2 + 11− x
)′=
((x2 + 11− x
) 13)′
=13
(x2 + 11− x
)− 23
·(
x2 + 11− x
)′=
13
(x2 + 11− x
)− 23
·(
2x · (1− x)− (−1)(x2 + 1)(1− x)2
)=
=13
(x2 + 11− x
)− 23
·(−x2 + 2x + 1
(1− x)2
)
:'d sirq
:ik lawpe millkd itl xefbp(3√
x +√
x
)′=((
x +√
x) 1
3)′
=13(x +
√x)− 2
3 ·(x +
√x)′ =
1
3 (x +√
x)23·(
1 +1
2√
x
)
:'e sirq
:ik lawpe millkd itl xefbp
(arcsin
(x√
1 + x2
))′=
(x√
1+x2
)′√
1−(√
x√1+x2
)2=
1 ·√
1 + x2 − 1·2x·x2·√
1+x2√1− x2
1+x2
=
1
(1+x2)32√
1− x2
1+x2
=
=1
(1 + x2)32 ·√
11+x2
:'f sirq
:ik lawpe millkd itl xefbp(1√
3x2 + 1
)′=((
3x2 + 1)− 1
2)′
= −12(3x2 + 1
)− 32 ·(3x2 + 1
)′= −1
21
(3x2 + 1)32· 6x =
=−3x
(3x2 + 1)32
14
ilaib aia`'` `"ecg
:'g sirq
:ik lawpe millkd itl xefbp
(arcsin(cos x))′ =1√
1− (cos x)2(cos x)′ =
sinx√1− (cos x)2
:2 dl`y
:'` sirq
g(x) = F (x)− h(x) ik lawp if` a dcewpa dxifb F (x) y dlilya gipp .dpekp dprhd.dxizq efe zexifb zeivwpet ly yxtdk a dcewpa dxifb
:'a sirq
a zexifb `l ody h(x) = −|x| e g(x) = |x| efk zicbp `nbec gwp .dpekp `l dprhd.R lka dxifb F (x) = |x| − |x| = 0 la` 0
:'b sirq
ik d`xp .h(x) = sin2(πx) e g(x) = [x] efk zicbp `nbec gwp .dpekp `l dprhdxy`k (m,m + 1) dxevdn rhw lka lk mcew .dxifb F (x) = [x] sin2(πx) dltknd
zecewpa zexifb wecap .zexifb zeivwpet ly dltknk dxifb F (x) f` m ∈ Z:dxcbd itl zenlyd
limx→m+
F (x)− F (m)x−m
= limx→m+
[x] sin2(πx)− 0x−m
= limx→m+
m sin2(π(x−m))x−m
=
limx→m+
sin2(πx)(π(x−m))2
· m(π(x−m))2
x−m= lim
x→m+m(x−m)π2 = 0
:ik lawp ote` eze`a
limx→m−
F (x)− F (m)x−m
= limx→m−
[x] sin2(πx)− 0x−m
= limx→m−
(m− 1) sin2(π(x−m))x−m
=
limx→m−
sin2(πx)(π(x−m))2
· (m− 1)(π(x−m))2
x−m= lim
x→m−(m− 1)(x−m)π2 = 0
.F ′(m) = 0 okle
:'c sirq
15
ilaib aia`'` `"ecg
a zexifb `l ody h(x) = |x| e g(x) = |x| efk zicbp `nbec gwp .dpekp `l dprhd.R lka dxifb F (x) = x2 la` 0
:3 dl`y
:'` sirq
:miiwzn dxcbdd itl if` dxifb f(x) m` .dpekp dprhd
f ′(x) = lim∆x→0
f(x0 + ∆x)− f(x0)∆x
okle n →∞ xy`k 1n → 0 mb f` ∆x → 0 xy`k ik xexae 1
n z` ∆x mewna aezkp:ik lawp
f ′(x) = limn→∞
f(x0 + 1
n
)− f(x0)
1n
= limn→∞
n
(f
(x0 +
1n
)− f(x0)
)
:'a sirq
idz , zicbp `nbec gwp .dpekp `l dprhd
f(x) =
1 , x ∈ Q
0 , x /∈ Q
mpne` iteq leabd xnelk limn→∞ n(f(x + 1
n
)− f(x)
)= 0 ik lawp jkl i`
.dxifb `l i`ceea okle dtivx `l divwpetd
:'b sirq
leabd zxcbde zetivxn okle ef dcewpa dtivx mb okle a a dxifb f(x)miiwzn |x− a| < δ miiwnd x lkly jk δ > 0 miiw ε > 0 lkly meyxl lkep
a ly daiaq yiy lawp ε = 1 gwip m` zxne` z`f .f(a)− ε < f(x) < f(a) + ε
.|f(x)| < max{|f(a) + 1|, |f(a)− 1|} day
:4 dl`y
:'` sirq
:ik lawp ϕ ly zetivxd jnq lre zxfbpd zxcbd itl
f ′(a) = limx→a
(x− a)ϕ(x)− 0x− a
= limx→a
ϕ(x) = ϕ(a)
:'a sirq
16
ilaib aia`'` `"ecg
:cxtpa aygp mrtd wx f ′(a) meiw z` wecap dxcbd itl aey
limx→a+
|x− a|ϕ(x)− 0x− a
= limx→a+
(x− a)ϕ(x)− 0x− a
= ϕ(a) 6= 0
limx→a−
|x− a|ϕ(x)− 0x− a
= limx→a−
−(x− a)ϕ(x)− 0x− a
= −ϕ(a) 6= 0
`ly iptn zniiw `l f ′(a) okle ϕ(a) 6= −ϕ(a) ik miiw `l limx→a|x−a|ϕ(x)−0
x−a okle.oey`x oinn zetivx i` zeidl dleki
:'b sirq
zeivwpet ly dltknk dtivx `id .∏n
i=1 |x− ai| lynl `id zywaend divwpetd.'a sirq itl 1 ≤ i ≤ n lkl ai zecewpa dxiyb `l `ide zetivx
:5 dl`y
s > 0 xear dxcbd itl ok`e dtivx f divwpetd s ikxr eli` xear mcew wecaps ≤ 0 xear zxne` z`f .miiw `l leabd zxg` limx→0 xs sin 1
x = 0 = f(0) ik lawpf izn wecap s > 0 xear zrk .dtivx `l `id ik dxifb `l geha f divwpetd
:ok`e dxcbd itl dxifb
limx→0
xs sin 1x − 0
x− 0= lim
x→0xs−1 sin
1x
= 0
.miiw `l `ed zxg` s > 1 xy`k
:6 dl`y
:ik lawp jkl i` f(x) = u(x)v(x) idz
ln (f(x)) = ln(u(x)v(x)
)= v(x) ln (u(x))
:ik lawpe mitb`d ipy z` xefbp
1f(x)
· f ′(x) = v′(x) ln (u(x)) + v(x) · 1u(x)
· u′(x)
okl
f ′(x) = f(x)[v′(x) ln (u(x)) + v(x) · u′(x)
u(x)
]= u(x)v(x)
[v′(x) ln (u(x)) + v(x) · u′(x)
u(x)
]
17
ilaib aia`'` `"ecg
il`ivpxticd oeaygd ly miiceqid mihtynd 2
dnxt htyn 2.1
zlawn f(x) m` .x0 zinipt dcewpa dxifbe (a, b) megza zxcbend divwpet f(x) idz.f ′(x0) = 0 if` x0 a meniqwnd e` menipind z`
:1 libxz
.(0, π) rhwa f(x) = sin x divwpetd ly meniqwn e`vn
:oexzt
xnelk f ′(x0) = 0 miiwzdl jixv if` (0, π) rhwa meniqwn zcewp zniiw m`.rhwa meniqwnd epid sin
(π2
)= 1 ok`e x = π
2 m` wxe m` dfe cos x = 0
liawn elld zecewpa wiynd xyidy df dnxt htyn ly ixhne`bd yextd [∗].x d xivl
lex htyn 2.2
.f(a) = f(b) miiwzne (a, b) a dxifbe [a, b] a dtivx [a, b] megza zxcbend divwpet f(x) idz.f ′(c) = 0 y jk c ∈ (a, b) zniiw if`
:ze`nbec
zpwqn z` miniwnd c ikxr lk z` e`vne oezpd rhwa lex htyn i`pz z` ewca:htynd
.htynd i`pz z` wecap .[2, 4] rhwd z`e f(x) = x2 − 6x + 8 divwpetd z` gwp [`]ok enk .[2, 4] rhwa hxtae x lkl dxifbe dtivx okle mepilet `id eply divwpetd
:miiwzn
f(2) = 22 − 6 · 2 + 8 = 0 , f(4) = 42 − 6 · 4 + 8 = 0 ⇒ f(2) = f(4)
f ′(x) = 2x− 6 la` .f ′(c) = 0 y jk c ∈ (2, 4) dcewp zniiw lex htyn itl jkl i`:okle
0 = f ′(c) = 2c− 6 ⇒ c = 3
18
ilaib aia`'` `"ecg
.htynd i`pz z` wecap .[
π2 , 3π
2
]rhwd z`e f(x) = cos x divwpetd z` gwp [a]
enk .[
π2 , 3π
2
]rhwa hxtae x lkl dxifbe dtivx okle zixhpnl` `id eply divwpetd
:miiwzn ok
f(π
2
)= 0 , f
(3π
2
)= 0 ⇒ f
(π
2
)= f
(3π
2
)
f ′(x) = sin x la` .f ′(c) = 0 y jk c ∈(
π2 , 3π
2
)dcewp zniiw lex htyn itl jkl i`
:okle0 = f ′(c) = sin c ⇒ c = π
.htynd i`pz z` wecap .[0, 4] rhwd z`e f(x) = 12x−
√x divwpetd z` gwp [b]
enk .[0, 4] rhwa hxtae x lkl dxifbe dtivx okle zixhpnl` `id eply divwpetd:miiwzn ok
f(0) =12· 0−
√0 = 0 , f(4) =
12· 4−
√4 = 0 ⇒ f(0) = f(4)
f ′(x) = 12 −
12√
xla` .f ′(c) = 0 y jk c ∈ (0, 4) dcewp zniiw lex htyn itl jkl i`
:okle
0 = f ′(c) =12− 1
2√
c⇒
√c = 1 ⇒ c = 1
:libxz
.miiynn miyxey 2 weica yi ex − x− 4 = 0 d`eeynl ik egiked
:oexzt
divwpetd zetivxn if` f(4) ≈ 7.98 < 0 mbe f(−4) ≈ 46 > 0 e f(0)− 3 < 0 y oeiknd`eeynl miyxey 3 miniw ik gipp .miiynn miyxey ipy zegtl mpyiy raep
x1 < x2 < x3 ik gipp zeillkd zlabd ilae d`eeynd iyxey x1, x2, x3 eidi xnelklex htyn itl okle xyid lk lr dxifbe dtivx f(x) .f(x1) = f(x2) = f(x3) xnelk
f ′(x) = ex − 1 la` f ′(c1) = f ′(c2) = 0 y jk c2 ∈ (x2, x3) e c1 ∈ (x1, x2) zeniiw.dxizq efe x = 0 wx `ed ef d`eeynl oexztde
'fpxbl htyn 2.3
y jk c ∈ (a, b) zniiw if` (a, b) a dxifbe [a, b] a dtivx divwpet f(x) idz
f ′(c) =f(b)− f(a)
b− a
19
ilaib aia`'` `"ecg
:ze`nbec
zpwqn z` miniwnd c ikxr lk z` e`vne oezpd rhwa fpxbl htyn i`pz z` ewca:htynd
.htynd i`pz z` wecap .[−4, 6] rhwd z`e f(x) = x2 + x divwpetd z` gwp [`]jkl i` .[−4, 6] rhwa hxtae x lkl dxifbe dtivx okle mepilet `id eply divwpetd
y jk c ∈ (−4, 6) dcewp zniiw fpxbl htyn itl
f ′(c) =f(6)− f(−4)
6− (−4)
:ik lawp lkd jqa jkl i` f(6) = 42 , f(−4) = 12 e f ′(x) = 2x + 1 la`
2c + 1 = f ′(c) =42− 12
10⇒ 2c + 1 = 3 ⇒ c = 1
divwpetd .htynd i`pz z` wecap .[1, 2] rhwd z`e f(x) = x3+1 divwpetd z` gwp [a]htyn itl jkl i` .[1, 2] rhwa hxtae x lkl dxifbe dtivx okle mepilet `id eply
y jk c ∈ (1, 2) dcewp zniiw fpxbl
f ′(c) =f(2)− f(1)
2− 1
:ik lawp lkd jqa jkl i` f(2) = 9 , f(1) = 2 e f ′(x) = 3x2 la`
3c2 = f ′(c) =9− 2
1⇒ 3c2 = 7 ⇒ c = ±
√73
.[1, 2] rhway dcigid `id ik c =√
73 `id ziteqd daeyzd jkl i`
: libxz
ik egikednan−1(b− a) < bn − an < nbn−1(b− a)
.b > a > 0 lkl
:oexzt
dcewp zniiw fpxbl htyn itl okle [a, b] rhwa opeazpe f(x) = xn divwpetd z` xicbp:y jk a < c < b
f ′(c) =f(b)− f(a)
b− a⇔ ncn−1 =
bn − an
b− a⇔ bn − an = (b− a)ncn−1
20
ilaib aia`'` `"ecg
:ik lawp okle an−1 < cn−1 < bn−1 ik raep a < c < b y dcaerdn
(b− a)nan−1 < (b− a)ncn−1 < (b− a)nbn−1
xnelk(b− a)nan−1 < bn − an < (b− a)nbn−1
iyew htyn 2.4
lkl g′(x) 6= 0 sqepae (a, b) rhwa zexifbe [a, b] rhwa zetivx zeivwpet f(x), g(x) eidi:y jk c ∈ (a, b) zniiw if` x ∈ (a, b)
f ′(c)g′(c)
=f(b)− f(a)g(b)− g(a)
lhitel htyn 2.5
.∞∞ , 00 dxevdn zeleab aygl minieqn mi`pza epl xyt`i df htyn
:htyn
:ik gipp .dnvr a l ile` hxt x = a zaiaqa zexifb zeivwpet f(x), g(x) eidi
.limx→a f(x) = limx→a g(x) = 0 [`]
.g′(x) 6= 0 miiwzn a zaiaqa x 6= a lkl [a]
.limx→af ′(x)g′(x) miiw [b]
.limx→af ′(x)g′(x) l deeye limx→a
f(x)g(x) leabd miiw if`
:dxrd
l (l`nyn wx e`) oinin wx zexcben zeivwpetdy dxwna mb miiwzn htynd.limx→a+
f(x)g(x) = limx→a+
f ′(x)g′(x) f`e a
:htyn
21
ilaib aia`'` `"ecg
:ik gipp .(α,∞) zaiaqa zexifb zeivwpet f(x), g(x) eidi
.limx→∞ f(x) = limx→∞ g(x) = 0 [`]
.g′(x) 6= 0 miiwzn x > α lkl [a]
.limx→∞f ′(x)g′(x) miiw [b]
.limx→∞f ′(x)g′(x) l deeye limx→∞
f(x)g(x) leabd miiw if`
: milibxz
:1 dl`y
: jxtd e` gkedegiked limx→a+ f(x) = limx→a− f(x) mbe (a, b) gezt rhwa dxifb divwpet f(x) idz
.f ′(c) = 0 y jk c ∈ (a, b) yiy
:2 dl`y
:mi`ad mipeieiy i`d z` exzt
.0 < a < b xear b−a1+b < ln 1+b
1+a < b−a1+a [`]
.| arctanx− arctan y| ≤ |x− y| [a]
:3 dl`y
lk oia f` , f ′(x)g(x) 6= f(x)g′(x) zeniiwne x lkl zexifb g(x) e f(x) m` ik egiked.g(x) ly yxey yi f(x) ly miyxey ipy
:4 dl`y
.edylk irah xtqn n idie (0, 1) a dxifbe [0, 1] a dtivx divwpet f(x) idz.f(1)− f(0) = f ′(c)
ncn−1 miiwnd 0 < c < 1 miiwy egiked
:5 dl`y
f(x0) = f(x1) = f(x2) = f(x3) y gippe (a, b) rhwa minrt 3 dxifbd divwpet f(x) idzx0 < c < x3 y jk c dcewp zniiwy egiked .a < x0 < x1 < x2 < x3 < b xy`k
.zqt`zn c dcewpa ziyilyd zxfbpd xnelk f (3)(c) = 0 zniiwnd
:6 dl`y
yxey xzeid lkl p(x) l yi (−∞,−1) rhwa ik egiked , p(x) = 3x4 + 4x3 + C idi
22
ilaib aia`'` `"ecg
.cg` iynn
: zepexzt
:1 dl`y.xgapy (a, b) rhw lka f(x) = x `nbecl ! dpekp `l of dprh
: d`ad dipal al eniy mpne`:`ad ote`a [a, b] rhwa g(x) divwpet xicbp
g(x) =
limt→a+ f(t) , x = a
f(x) , x ∈ (a, b)
limt→a− f(t) , x = b
itl jkl i` .g(a) = g(b) miiwzn mbe (a, b) rhwa dxifbe [a, b] rhwa dtivx g(x)miiwzne c ∈ (a, b) y oeik .g′(c) = 0 day c ∈ (a, b) dcewp zniiw lex htyn
.f ′(c) = g′(c) = 0 ik xexa f` x ∈ (a, b) lkl f(x) = g(x)
x = b a zetivx i` zcewp yie okzi la` oekp hrnk df ok m`! z`f zeyrl ozip `l f`e
:2 dl`y
:'` sirq
okle (a, b) a dxifbe [a, b] a dtivx divwpet f(x) ik xexa f` f(x) = ln(1 + x) idz:miiwzny jk c ∈ (a, b) zniiw fpxbl htyn itl
f ′(c) =1
1 + c=
f(b)− f(a)b− a
=ln(1 + b)− ln(1 + a)
b− a
:ik lawp minzixbel iweg itl
ln(1 + b)− ln(1 + a) = ln(
1 + b
1 + a
)
:ik eplaiw okl
ln(
1+b1+a
)b− a
=1
1 + c
:ik lawpe b− a a oeieiyd z` letkp
ln(
1 + b
1 + a
)=
b− a
1 + c
23
ilaib aia`'` `"ecg
:ik lawp f` a < c < b e zeid
b− a
1 + b<
b− a
1 + c<
b− a
1 + a
:ik eplaw lkd jqa okl
b− a
1 + b< ln
(1 + b
1 + a
)<
b− a
1 + a
:'a sirq
`ll gippe , x 6= y eay dxwna lthp .oeieiy miiwzn x = y eay dxwnay xexa:jk oeieiy i`d z` azkyl ozip .x > y ik zeillkd zlabd∣∣∣∣arctanx− arctan y
x− y
∣∣∣∣ ≤ 1
dxifbe df rhwa dtivx divwpetd , [x, y] rhwa f(t) = arctan(t) divwpeta opeazpcvn f ′(c) = arctan x−arctan y
x−y exeary c ∈ (x, y) miiw fpxbl htyn itl okle (x, y) rhwa:ik milawn okle .f ′(c) = 1
1+c2 y xxeby dn f ′(t) = 11+t2 ipy∣∣∣∣arctanx− arctan y
x− y
∣∣∣∣ ≤ 1 ⇒∣∣∣∣ 11 + c2
∣∣∣∣ ≤ 1
.c lkl miiwzn dfe
:3 dl`y
:ik lawpe (dpnd llk itl) h(x) z` xefbp h(x) = f(x)g(x) xicbp
h′(x) =f ′(x)g(x)− f(x)g′(x)
(g(x))26=
given
0
x1, x2 eidi xnelk , f(x) iyxey 2 lk oia g(x) ly yxey miiw `l ik dlilya gipp:miiwzn xnelk f(x) iyxey
f(x1) = f(x2) = 0 ⇒ h(x1) = h(x2) = 0
miniiwzn htynd i`pz , (x1 < x2 k"da) [x1, x2] rhwd xear lex htyna ynzyp.zqt`zn dpi` h′(x) ik dxizq ef la` h′(c) = 0 y jk c ∈ (x1, x2) zniiw okl rhwa
:4 dl`y
24
ilaib aia`'` `"ecg
mbe (0, 1) rhwa dxifb mbe [0, 1] rhwa dtivx gn(x) .n > 0 xear gn(x) = xn xicbpdcewp zniiw iyew htyn itl okl .g′n(x) = nxn−1 6= 0 miiwzn x ∈ (0, 1) lkl
:miiwzn day c ∈ (0, 1)
f(a)− f(0)gn(1)− gn(0)
=f ′(c)g′n(c)
=f ′(c)ncn−1
xnelk
f(1)− f(0) =f ′(c)ncn−1
(gn(1)− gn(0)) =f ′(c)ncn−1
(1− 0) =f ′(c)ncn−1
:5 dl`y
okle a < x0 < x1 < x2 < x3 < b xy`k (a, b) rhwa minrt 3 dxifb f(x)mirhwa lex htyn i`pz miniiwzn okl .[x0, x1], [x1, x2], [x2, x3] mirhwa dtivxy jk c1 ∈ (x0, x1), c2 ∈ (x1, x2), c3 ∈ (x2, x3) zeniw jkl i` [x0, x1], [x1, x2], [x2, x3]
mirhwa f ′(x) divwpetd xear lex htyn z` aey lirtp .f ′(c1) = f ′(c2) = f ′(c3) = 0.f ′′(c4) = f ′′(c5) = 0 y jk c4 ∈ (c1, c2), c5 ∈ (c2, c3) zecewp zeniw okle [c1, c2], [c2, c3]
okle [c4, c5] rhwa f ′′(x) divwpetd xear lex htyn i`pz miniiwzn jkl i`.c6 = c ∈ (x0, x3) ik xexae c = c6 idze f (3)(c6) = 0 y jk c6 ∈ (c4, c5) zniiw
:6 dl`y
if` p(a) = p(b) = 0 e a < b y jk a, b eidi , miyxey 2 yi p(x) l ik dlilya gipp:y jk c ∈ (a, b) zniiw okle lex htyn i`pz miniwzn
p′(c) = 12c3 + 12c2 = 0 ⇒ 12c2(c + 1) = 0 ⇒ c = 0 or c + 1 = 0
yi p(x) l okl .dxizq efe (−∞,−1) l zkiiy `l c la` , c = −1 f` c 6= 0 y oeikn.(−∞,−1) a cg` iynn yxey xzeid lkl
:htyn
:ik gipp .dnvr a l ile` hxt x = a zaiaqa zexifb zeivwpet f(x), g(x) eidi
.limx→a f(x) = limx→a g(x) = ∞ [`]
.g′(x) 6= 0 miiwzn a zaiaqa x 6= a lkl [a]
.limx→af ′(x)g′(x) miiw [b]
.limx→af ′(x)g′(x) l deeye limx→a
f(x)g(x) leabd miiw if`
.zeiccv cg od divwpetd zexcbd xy`k oekp htynd el`d mixwna mb [∗]
25
ilaib aia`'` `"ecg
:dpwqn
:y jk a dcewpa minrt n zexifb zeivwpet f(x), g(x) eidi
f(a) = f ′(a) = · · · = f (n−1)(a) = 0
g(a) = g′(a) = · · · = g(n−1)(a) = 0
.∞∞ ly dxwna mb .limx→af(x)g(x) l deey `ed if` limx→a
f(n)(x)g(n)(x)
leabd miiw m`
:1 libxz
.α > 0 e a > 1 xy`k limx→∞xα
ax leabd z` eayg
:oexzt
z` lhitel itl aygp okle x lkl qt`n dpey dpknd zxfbp ∞∞ dxevdn leab edf
:ik lawpe zexfbpd zpn
limx→∞
αxα−1
ax ln a=(∞∞
):ik lawpe lhitel aey dyrp okle
limx→∞
(α)(α− 1)xα−2
ax(ln a)2=(∞∞
)
:ik lawp f`e α− k < 0 y cr minrt k xefbl jiynp okle
limx→∞
(α)(α− 1) · · · (α− k)xα−k−1
ax(ln a)k=(
0∞
)= 0
:2 libxz
.limx→0
(1x −
1sin x
)leabd z` eayg
:oexzt
ynzyp okle limx→0
(sin x−xx sin x
)=(
00
)dxevdn df ik mii`ex ep` f`e leabd z` hytp
:ik lawpe lhitela
limx→0
(sinx− x
x sinx
)=(
00
)=︸︷︷︸
L′Hospital
limx→0
(cos x− 1
sinx + x cos x
)=(
00
)=︸︷︷︸
L′Hospital
limx→0
(− sinx
cos x + cos x− x sinx
)=
02
= 0
26
ilaib aia`'` `"ecg
:3 libxz
.limx→∞ x(
π2 − arctanx
)leabd z` eayg
:oexzt
okle limx→∞
(π2−arctan x
1x
)=(
00
)dxevdn df ik mii`ex ep` f`e leabd z` hytp
:ik lawpe lhitela ynzyp
limx→∞
π2 − arctanx
1x
=(
00
)=︸︷︷︸
L′Hospital
limx→∞
− 11+x2
− 1x2
= limx→∞
x2
1 + x2= 1
:dxrd
leab lr ef jxca melk cibdl ozip `l f` miiw `l zexfbpd zpn ly leabd m`.onvr zeivwpetd zpn
:milibxz
:mi`ad zeleabd z` eayg
limx→0+2ex2
−2cos(x)ln(x) [a] limx→1
x2−2x+12x2−x−1 [`]
limx→0+ sin(x) · ln(x) [c] limx→0 (tanx)2 sin(x) [b]
limx→∞xln x
(ln x)x [e] limx→0+
(1− xxx)
[d]
limx→0
(sin x
x
) 11−cos x [g] limx→0
(cos xcos 2x
) 1x2 [f]
limx→0ex−e−x−2x
x−sin x [i] limx→0
(1x2 − cot2 x
)[h]
:zeaeyz
:ik lawpe lhitel htyna ynzyp [`]
limx→1
x2 − 2x + 12x2 − x− 1
=(
00
)=︸︷︷︸
L′Hospital
limx→1
2x− 24x− 1
=03
= 0
27
ilaib aia`'` `"ecg
:lhitela ynzydl jxev oi`e heyt leab edf [a]
limx→0+
2ex2 − 2cos(x)ln(x)
= 0
:ik lawp okle eln x = x y dcaera ynzyp [b]
limx→0
(tanx)2 sin(x) = limx→0
eln(tan x)2 sin(x)= elimx→0 2 sin(x) ln(tan(x))
:ik lawpe jixrnd ly leabd z` aygp zrk
limx→0
2 sin(x) ln(tan(x)) = 2 limx→0
ln(tan(x))1
sin(x)
=(−∞∞
)=︸︷︷︸
L′Hospital
= −2 limx→0
1tan(x) cos2(x)
cos(x)sin2(x)
= −2 limx→0
sin2(x)tan(x) cos3(x)
= −2 limx→0
sin2(x)sin(x)cos(x) · cos3(x)
=
= −2 limx→0
sin(x)cos2(x)
= 0
:ik lawp ixewnd leabl dxfga jkl i`
limx→0
(tanx)2 sin(x) = elimx→0 2 sin(x) ln(tan(x)) = limx→0
e0 = 1
:ik lawpe lhitel htyna ynzyp [c]
limx→0+
sin(x) · ln(x) = limx→0+
ln(x)1
sin(x)
=(−∞∞
)=︸︷︷︸
L′Hospital
= limx→0+
1x
− cos(x)sin2(x)
=
= − limx→0+
sin2(x)x cos(x)
= − limx→0+
sin(x)x
· sin(x)cos(x)
= −1 · limx→0+
tan(x) = −1 · 0 = 0
:ik lawpe g(x) = xx onqp [d]
limx→0+
g(x) = limx→0+
xx = limx→0+
eln(xx) = limx→0+
ex ln(x)
:`ad leaba lthp jkl i`
limx→0+
x ln(x) = limx→0+
ln(x)1x
=(∞∞
)=︸︷︷︸
L′Hospital
= limx→0+
−1x1x2
= limx→0+
−x = 0
:ik lawp oklelim
x→0+g(x) = lim
x→0+ex ln(x) = e0 = 1
28
ilaib aia`'` `"ecg
limx→0+
xg(x) = limx→0+
xxx
= limx→0+
eg(x) ln(x) = e1·(−∞) = 0
:ik lawpe ixewnd leabl xefgp
limx→0+
(1− xxx
)= 1− 0 = 1
:ik lawp okle eln x = x y dcaera ynzyp [e]
limx→∞
xln x
(lnx)x= lim
x→∞
eln x·ln x
eln(ln x)·x = limx→∞
eln2 x−x ln(ln(x)) = elimx→∞ ln2 x−x ln(ln(x))
:leabd lr lkzqp zrk
limx→∞
ln2 x− x ln(ln(x)) = limx→∞
x
(ln2 x
x− x ln(ln(x))
)= −∞
:ik
limx→∞
ln2 x
x=(∞∞
)=︸︷︷︸
L′Hospital
limx→∞
2 ln · 1x1
= limx→∞
2 ln x
x=(∞∞
)=︸︷︷︸
L′Hospital
limx→∞
2x
1= 0
:ik lawp okle eln x = x y dcaera ynzyp [f]
limx→0
( cos x
cos 2x
) 1x2
= limx→0
e1
x2 ln( cos xcos 2x ) = lim
x→0e
ln(cos x)−ln(cos 2x)x2 = e
32
:ik
limx→0
ln(cos x)− ln(cos 2x)x2
=(
00
)=︸︷︷︸
L′Hospital
limx→0
2 sin 2xcos 2x − sin x
cos x
2x= lim
x→0
tan 2x− tanx
2x=
(00
)=︸︷︷︸
L′Hospital
limx→0
4cos2 2x −
1cos2 x
2=
32
:ik lawp okle eln x = x y dcaera ynzyp [g]
limx→0
(sinx
x
) 11−cos x
= limx→0
eln( sin x
x )1−cos x = e−
13
:ik
limx→0
ln(
sin xx
)1− cos x
=(
00
)=︸︷︷︸
L′Hospital
limx→0
xsin x ·
x cos x−sin xx2
sinx= lim
x→0
x cos x− sinx
x sin2 x=
29
ilaib aia`'` `"ecg
(00
)=︸︷︷︸
L′Hospital
limx→0
cos x− x sinx− cos x
sin2 x + 2x sinx cos x= lim
x→0
−x
sinx + 2x cos x=
(00
)=︸︷︷︸
L′Hospital
limx→0
=−1
3 cos x− 2x sinx= −1
3
:`ad gezitd z` d`xp [h]
limx→0
(1x2− cot2 x
)= lim
x→0
sin2 x− x2 cos2 x
x2 sin2 x= lim
x→0
sin2 x− x2(1− sin2 x)x2 sin2 x
=
limx→0
sin2 x− x2
x2 sin2 x+ 1 = lim
x→0
sin2 xx2 − 1sin2 x
+ 1 =23
:ik
limx→0
sin2 xx2 − 1sin2 x
=(
00
)=︸︷︷︸
L′Hospital
limx→0
2x2 sin x cos x−2x sin2 xx4
2 sinx cos x= lim
x→0
x cos x− sinx
x3 cos x=
limx→0
x− tanx
x3=(
00
)=︸︷︷︸
L′Hospital
limx→0
1− 1cos2 x
3x2=(
00
)=︸︷︷︸
L′Hospital
limx→0
−2 sin x cos xcos4 x
6x=
limx→0
− sinx
3x cos3 x=(
00
)=︸︷︷︸
L′Hospital
limx→0
=− cos x
3 cos3 x− 9x cos2 x sinx= −1
3
:ik lawpe lhitel htyna ynzyp [i]
limx→0
ex − e−x − 2x
x− sinx=(
00
)=︸︷︷︸
L′Hospital
limx→0
ex + e−x − 21− cos x
=(
00
)=︸︷︷︸
L′Hospital
limx→0
ex − e−x
sinx=(
00
)=︸︷︷︸
L′Hospital
limx→0
ex + e−x
cos x= 2
xeliih zgqep 3
idi :xfr zprh `iap htynd iptle mepilet zxfra divwpet ly aexw epid illkd oeirxd.P (k)(a) = ak · k! f` k ≤ n idie n dlrnn mepilet P (x) = a0 + a1(x− a) + · · ·+ an(x− a)n
:xeliih zgqep
daiaqa idylk dcewp x idze x = x0 dcewpd zaiaqa minrt n + 1 dxifb f(x) idz:`ed f(x) xear mepiletd if`
f(x) = f(x0) +f ′(x0)
1!(x− x0) +
f ′′(x0)2!
(x− x0)2 + · · ·+ f (n)(x0)n!
(x− x0)n
30
ilaib aia`'` `"ecg
:dpid fpxbl itl zix`yd zgqep xy`k
Rn(x) =f (n+1)(c)(n + 1)!
(x− x0)n+1 , c ∈ (x0, x)
:dxrd
:jk zi`xp `ide oxelwn z`gqep z`xwp xeliih zgqep if` x0 = 0 xy`k
f(x) = f(0) +f ′(0)
1!· x +
f ′′(0)2!
· x2 + · · ·+ f (n)(0)n!
· xn + Rn(x)
:1 libxz
.f(x) = ln(x + 1) ly oxelwn zgqep z` e`vn
:oexzt
:ok`e k xcq cr divwpetd ly zexfbpd z` aygp
f(x) = ln(x + 1) ⇒ f ′(x) =1
x + 1= (x + 1)−1 ⇒ f ′′(x) = −(x + 1)−2 ⇒
f (3)(x) = 2(x + 1)−3 ⇒ f (4)(x) = −6(x + 1)−4
a aygp xy`k okle f (k)(x) = (−1)k+1(k − 1)!(x + 1)−k ik divwecpi`a gikedl ozip:ik lawp x0 = 0
f(x) = f(0) +f ′(0)
1!(x) +
f ′′(0)2!
(x)2 + · · ·+ f (n)(0)n!
(x)n + Rn(x) =
= 0 +(−1)2 · 0!
1!· x +
(−1)3 · 1!2!
· x2 + · · ·+ (−1)k+1 · (k − 1)!k!
· xk + Rk(x) =
x− 12x2 +
13x3 + · · ·+ (−1)k+1
kxk + Rk(x)
.c ∈ (0, x) xear Rk(x) =((−1)k+2k!(c + 1)−k−1
)xk+1
(k+1)! `id d`ibyd xy`k
:2 libxz
.10−2 = 0.01 lr dlrz `l d`ibyd xy`k ln(1.2) z` eayg
31
ilaib aia`'` `"ecg
:oexzt
x = 0.2 gwp m` .|Rk(x)| < 0.01 miiwi fpxbl itl d`ibyd xai`y jk k miytgn
xy`k f(k+1)(c)(k+1)! · xk+1 jk d`xp d`ibyd xai`e ln(1.2) = ln(1 + x) ik lawp if`
:ik lawp ln(1 + x) xear eply dxwna .c ∈ (0, 0.2)
|Rk(x)| =∣∣∣∣f (k+1)(c)
(k + 1)!· xk+1
∣∣∣∣ = ∣∣∣∣ (−1)k+1 · (k)!(k + 1)!
· (0.2)k+1
∣∣∣∣ < 1k + 1
· (0.2)k+1 =(0.2)k+1
k + 1
:ok`e z`f miiwn k dfi` wecap okle (0.2)k+1
k+1 < 0.01 ik yexcp okle
k = 1 :(0.2)2
2= 0.02 > 0.01
k = 2 :(0.2)3
3= 0.0004 < 0.01
:ik lawp okle k = 2 xear miiwzn df xnelk
f(x) ≈ P2(x) = 0 + x− 12x2 = x− 1
2x2
.ln(1.2) = 0.1823 ok`e P2(0.2) = (0.2)− 12 (0.2)2 = 0.18 okle
milibxz:1 dl`y
:ze`ad zeivwpetd z` xelih xehl egzt.x0 = 0 aiaq f(x) = cos(x) [`]
.x0 = 1 aiaq f(x) = 1x [a]
.x0 = 2 aiaq f(x) = ln(x) [b]
:2 dl`y
lcebl dkxrd epze sinx ly 3 xcqn oxelwn xeh zxfra sin 3◦ z` eayg [`].d`ibyd
, dcewpd ixg` zextq 3 ly weica , ex ly oxelwn xeh zxfra e z` eayg [a].0.5 · 10−3 lr dlrz `l d`ibydy zxne` z`f
:1 dl`y
32
ilaib aia`'` `"ecg
:'` sirq
:ok`e x0 = 0 a odly jxrd z`e n xcq cr divwpetd ly zexfbp `vnp
f(x) = cos(x) ⇒ f(0) = 1f ′(x) = − sin(x) ⇒ f ′(0) = 0f ′′(x) = − cos(x) ⇒ f ′′(0) = −1f (3)(x) = sin(x) ⇒ f (3)(0) = 0f (4)(x) = cos(x) ⇒ f (4)(0) = 1
...f (n)(x) = cos
(x + nπ
2
)⇒ f (n)(0) = cos
(nπ2
):ik lawp xelih zgqep itl jkl i`
f(x) = 1− x2
2!+
x4
4!− x6
6!+ · · ·+ (−1)nx2n
(2n)!+ Rn(x)
.c ∈ (0, x) xear Rn(x) = cos(c + (n+1)π
2
)xn+1
(n+1)! y jk
:'a sirq
:ok`e x0 = 1 a odly jxrd z`e n xcq cr divwpetd ly zexfbp `vnp
f(x) = 1x = x−1 ⇒ f(1) = 1 = (−1)0 · 0!
f ′(x) = −x−2 ⇒ f ′(1) = −1 = (−1)1 · 1!f ′′(x) = 2x−3 ⇒ f ′′(1) = 2 = (−1)2 · 2!f (3)(x) = −6x−4 ⇒ f (3)(1) = −6 = (−1)3 · 3!
...f (n)(x) = (−1)nn!x−(n+1) ⇒ f (n)(1) = (−1)nn!
:ik lawp xelih zgqep itl jkl i`
f(x) = f(1) +f ′(1)
1!(x− 1) +
f ′′(1)2!
(x− 1)2 + · · ·+ f (n)(1)n!
(x− 1)n + Rn(x) =
= 1 + (−1)1(x− 1) + (−1)2(x− 1)2 + · · ·+ (−1)n(x− 1)n + Rn(x) = 1 +n∑
k=1
(−1)k(x− 1)k + Rn(x)
.c ∈ (1, x) xear Rn(x) = (1−x)n+1
cn+2 y jk
:'b sirq
33
ilaib aia`'` `"ecg
:ok`e x0 = 2 a odly jxrd z`e n xcq cr divwpetd ly zexfbp `vnp
f(x) = ln(x) ⇒ f(2) = ln 2f ′(x) = 1
x ⇒ f ′(2) = 12 = (−1)0 · 0!
21
f ′′(x) = −x−2 ⇒ f ′′(2) = − 14 = (−1)1 · 1!
22
f (3)(x) = 2x−3 ⇒ f (3)(2) = 14 = (−1)2 · 2!
23
f (4)(x) = −6x−4 ⇒ f (4)(2) = − 38 = (−1)3 · 4!
24
...f (n)(x) = (−1)n−1(n− 1)!x−n ⇒ f (n)(2) = (−1)n−1 (n−1)!
2n
:ik lawp xelih zgqep itl jkl i`
f(x) = f(2) +f ′(2)
1!(x− 2) +
f ′′(2)2!
(x− 2)2 + · · ·+ f (n)(2)n!
(x− 2)n + Rn(x) =
= ln 2 + (−1)0(x− 2)1 · 21
+ (−1)1(x− 2)2
2 · 22+ (−1)2
(x− 2)3
3 · 23+ · · ·+ (−1)n−1 (x− 2)3
n · 2n+ Rn(x) =
ln 2 +n∑
k=1
(−1)k−1 (x− 2)3
k · 2k+ Rn(x)
.c ∈ (2, x) xear Rn(x) = (−1)n
n+1
(x−2
c
)n+1 y jk
:2 dl`y
:'` sirq
`id d`ibyde P3(x) = x− x3
3! ik lawp 3 xcq cr oxelwn gezit it lr:d`ibyd z` jixrpe R3(x) = sin(c)x4
4!∣∣∣R3
( π
60
)∣∣∣ = ∣∣∣∣ sin(c)4!
( π
60
)4∣∣∣∣ ≤ π4
24 · 604≈ 0.3 · 10−6
lawpe aygp zrk .dcewpd ixg` zewiecn zextq 5 lra didi P3
(π60
)zxne` z`f
:ik
sin 3◦ ≈ P3
( π
60
)=
π
60− π3
6 · 603≈ 0.05234
:'a sirq
:dpid ex ly oixelwn zgqep
ex = 1 + x +x2
2!+
x3
3!+ · · ·+ xn
n!+ Rn(x) =
∞∑n=0
xn
n!
34
ilaib aia`'` `"ecg
:d`ibyd ly mqg `vnp .c ∈ (0, x) xear Rn(x) = ecxn+1
(n+1)! y jk
Rn(1) =ec
(n + 1)!<
e
(n + 1)!<
3(n + 1)!
lr zrk .e < 3 mby oaenke dler divwpet ex ik e0 < ec < e1 f` 0 < c < 1 y oeik3
(7+1)! ≈ 0.074 · 10−3 < 0.5 · 10−3 miiwny oey`xd `ed n = 7 ik mi`ven dirhe ieqip ici:z` aygp okle
P7(1) =7∑
n=0
1n!≈ 2.718
35
