Нэг хувьсагчийн функц

Íýã
õóâüñàã÷èéí
ôóíêö,
ò³³íèéã
èëýðõèéëýõ
àðãóóä
Ôóíêöèéí
õÿçãààð
Íýã õóâüñàã÷èéí ôóíêö
Ä.Áàòò°ð
email: batturdn@gmail.com
Óëààíáààòàð Èõ Ñóðãóóëü
www.ulaanbatar.edu.mn
Ä.Áàòò°ð email: batturdn@gmail.com Ìàòåìàòèê 1

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
1 Íýã õóâüñàã÷èéí ôóíêö, ò³³íèéã èëýðõèéëýõ àðãóóä
Óðâóó ôóíêö
“íäñýí ýëåìåíòàð ôóíêö³³ä
Ýëåìåíòàð ôóíêö³³ä
2 Ôóíêöèéí õÿçãààð
f(x) ôóíêöèéí x → x0 ³åèéí õÿçãààð
x → ∞, x → +∞, x → −∞ ³åèéí õÿçãààð

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Íýã õóâüñàã÷èéí ôóíêöÍýã õóâüñàã÷èéí ôóíêöèéí ³íäýñ
X, Y òîîí îëîíëîãóóä ãýå. Õýðýâ X x ýëåìåíò á³ðò òîäîðõîé
ä³ðìýýð Y y öîð ãàíö ýëåìåíò õàðãàëçñàí áàéâàë X îëîíëîã
äýýð òîäîðõîéëîãäñîí, Y îëîíëîãîîñ óòãàà àâäàã, ôóíêö
°ã°ãäñ°í áàéíà ãýýä

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
y = f(x), y = g(x),

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
y = f(x), y = g(x), f : x → y,

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
y = f(x), y = g(x), f : x → y, X
f
−→ Y ã.ì.
òýìäýãëýí áè÷íý.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
y = f(x), y = g(x), f : x → y, X
f
−→ Y ã.ì.
Õóâüñàõ õýìæèãäýõ³³í x:
Õóâüñàõ õýìæèãäýõ³³í y:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
y = f(x), y = g(x), f : x → y, X
f
−→ Y ã.ì.
Õóâüñàõ õýìæèãäýõ³³í x: “ë õàìààðàõ õóâüñàã÷ (Àðãóìåíò)
Õóâüñàõ õýìæèãäýõ³³í y:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
y = f(x), y = g(x), f : x → y, X
f
−→ Y ã.ì.
Õóâüñàõ õýìæèãäýõ³³í y: Õàìààðàõ õóâüñàã÷ (Ôóíêö)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
y = f(x), y = g(x), f : x → y, X
f
−→ Y ã.ì.
y0 = f(x0) (y0 = f(x)|x=x0
) òîî íü y = f(x) ôóíêöèéí
x = x0 öýã äýýðõ óòãà.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
y = f(x), y = g(x), f : x → y, X
f
−→ Y ã.ì.
y0 = f(x0) (y0 = f(x)|x=x0
) òîî íü y = f(x) ôóíêöèéí
x = x0 öýã äýýðõ óòãà.
òîäîðõîéëîãäîõ ìóæ
X∼D(f)
f
−→ óòãûí ìóæ
Y ∼E(f)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Íýã õóâüñàã÷èéí ôóíêöÍýã õóâüñàã÷èéí ôóíêöèéã èëýðõèéëýõ àðãóóä
Òîìú¼îíû àðãà:
Ìàòåìàòèêèéí òîìú¼îãîîð ôóíêöèéã èëýðõèéëýõ
àðãà

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
àðãà
y = x2
−∞ ≤ x ≤ ∞
y = x2
+ 2x + 3, 0 ≤ x ≤ 1
y =
√
1 − x2, −1 ≤ x ≤ 1.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
àðãà
y = x2
−∞ ≤ x ≤ ∞
y = x2
+ 2x + 3, 0 ≤ x ≤ 1
y =
√
1 − x2, −1 ≤ x ≤ 1.
Õ³ñíýãòèéí àðãà:
Õóâüñàõ õýìæèãäýõ³³í x-èéí òîäîðõîé õýäýí øèðõýã
óòãà òóñ á³ðò õóâüñàõ õýìæèãäýõ³³í y-èéí íýã óòãûã
õàðãàëçóóëàí õ³ñíýãòýýð èëýðõèéëýõ àðãà

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
àðãà
y = x2
−∞ ≤ x ≤ ∞
y = x2
+ 2x + 3, 0 ≤ x ≤ 1
y =
√
1 − x2, −1 ≤ x ≤ 1.
Õ³ñíýãòèéí àðãà:
Õóâüñàõ õýìæèãäýõ³³í x-èéí òîäîðõîé õýäýí øèðõýã
óòãà òóñ á³ðò õóâüñàõ õýìæèãäýõ³³í y-èéí íýã óòãûã
õàðãàëçóóëàí õ³ñíýãòýýð èëýðõèéëýõ àðãà
x -4 -3 -2 -1 0 1 2 3 4
y 16 9 4 1 0 1 4 9 16
x 0 0,1 0,2 3 0,6 4 0,8
y -1 10 1 -2 8 0,5 -2

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Ãðàôèêèéí àðãà:
Êîîðäèíàòûí õàâòãàéí M(x, f(x)), (x ∈ X)
öýã³³äèéí îëîíëîãîîð ä³ðñëýãäýõ øóãàìûã
y = f(x) ôóíêöèéí ãðàôèê ãýíý

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Ãðàôèêèéí àðãà:
Êîîðäèíàòûí õàâòãàéí M(x, f(x)), (x ∈ X)
öýã³³äèéí îëîíëîãîîð ä³ðñëýãäýõ øóãàìûã
y = f(x) ôóíêöèéí ãðàôèê ãýíý
Çóðàã: Ôóíêöèéí ãðàôèê

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Íýã õóâüñàã÷èéí ôóíêöÔóíêö³³ä äýýðõ ³éëäë³³ä
• f áà g ôóíêö³³ä íü X îëîíëîã äýýð òîäîðõîéëîãäñîí áàéã.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Íèéëáýð áà ÿëãàâàð:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Íèéëáýð áà ÿëãàâàð: (f ± g)(x) = f(x) ± g(x)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
“ðæâýð:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
“ðæâýð: (f · g)(x) = f(x) · g(x)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
“ðæâýð: (f · g)(x) = f(x) · g(x)
Òîãòìîë òîîãîîð ³ðæèõ:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
“ðæâýð: (f · g)(x) = f(x) · g(x)
Òîãòìîë òîîãîîð ³ðæèõ: α = const, (α · f)(x) = αf(x)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
“ðæâýð: (f · g)(x) = f(x) · g(x)
Íîîãäâîð:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
“ðæâýð: (f · g)(x) = f(x) · g(x)
Íîîãäâîð:
f
g (x) = f(x)
g(x) , g(x) = 0

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
“ðæâýð: (f · g)(x) = f(x) · g(x)
Íîîãäâîð:
f
g (x) = f(x)
g(x) , g(x) = 0
Äàâõàð ôóíêö (äàâõàðëàëò):

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
“ðæâýð: (f · g)(x) = f(x) · g(x)
Íîîãäâîð:
f
g (x) = f(x)
g(x) , g(x) = 0
Äàâõàð ôóíêö (äàâõàðëàëò): Õýðýâ y = f(u) ôóíêöèéí
àðãóìåíò u íü áàñ ÿìàð íýã x õóâüñàã÷ààñ õàìààðñàí
u = φ(x) ôóíêö áàéâàë ýíý õýëõýýãýýð u = φ(x)
ôóíêöèéí óòãûí ìóæ äýýð òîäîðõîéëîãäîõ äàâõàð
ôóíêö:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
“ðæâýð: (f · g)(x) = f(x) · g(x)
Íîîãäâîð:
f
g (x) = f(x)
g(x) , g(x) = 0
Äàâõàð ôóíêö (äàâõàðëàëò): Õýðýâ y = f(u) ôóíêöèéí
àðãóìåíò u íü áàñ ÿìàð íýã x õóâüñàã÷ààñ õàìààðñàí
u = φ(x) ôóíêö áàéâàë ýíý õýëõýýãýýð u = φ(x)
ôóíêöèéí óòãûí ìóæ äýýð òîäîðõîéëîãäîõ äàâõàð
ôóíêö:
y = f[φ(x)] = F(x)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Íýã õóâüñàã÷èéí ôóíêöÔóíêöèéí çàðèì °â°ðì°ö ÷àíàðóóä
1. Òîãòìîë ôóíêö:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
1. Òîãòìîë ôóíêö: ∀x ∈ X õóâüä f(x) = c, c = const

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
2. Çààãëàãäñàí ôóíêö:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
X
f
−→ Y ôóíêöèéí õóâüä f(x) ≤ M, (f(x) ≥ m), áàéõ
òîãòìîë M(m) îëäîæ áàéâàë X îëîíëîã äýýð f(x)
ôóíêö íü äýýðýýñýý M òîîãîîð (äîîðîîñîî m òîîãîîð)
çààãëàãäñàí ôóíêö ãýíý.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
X
f
Äýýðýýñýý áà äîîðîîñîî íýãýí çýðýã çààãëàãäñàí áàéâàë
f(x) ôóíêöèéã X îëîíëîã äýýð çààãëàãäñàí ôóíêö ãýíý.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
X
f
3. Ìîíîòîí ôóíêö:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
X
f
3. Ìîíîòîí ôóíêö: X = (a, b) äýýð y = f(x) ôóíêö àâ÷ ³çüå.
Õýðýâ ∀x1, x2 ∈ X óòãóóäàä:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
X
f
x1 < x2 ⇒ f(x1) < f(x2) áîë f(x) °ñí°

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
X
f
x1 < x2 ⇒ f(x1) < f(x2) áîë f(x) °ñí°
x1 < x2 ⇒ f(x1) > f(x2) áîë f(x) áóóðíà

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
X
f
x1 < x2 ⇒ f(x1) < f(x2) áîë f(x) °ñí°
x1 < x2 ⇒ f(x1) ≤ f(x2) áîë f(x) ³ë áóóðíà

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
X
f
x1 < x2 ⇒ f(x1) < f(x2) áîë f(x) °ñí°
x1 < x2 ⇒ f(x1) ≤ f(x2) áîë f(x) ³ë áóóðíà
x1 < x2 ⇒ f(x2) ≥ f(x2) áîë f(x) ³ë °ñí°

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
4. Òýãø, ñîíäãîé ôóíêö:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
4. Òýãø, ñîíäãîé ôóíêö: Òýãø õýìòýé X = (−a, a) çàâñàðò
òîäîðõîéëîãäñîí y = f(x) ôóíêöèéí õóâüä:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Õýðýâ f(−x) = f(x) áîë f(x) íü òýãø ôóíêö

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
òýãø ôóíêöèéí ãðàôèê OY òýíõëýãò òýãø õýìòýé

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Õýðýâ f(−x) = −f(x) áîë f(x) íü ñîíäãîé ôóíêö

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
ñîíäãîé ôóíêöèéí ãðàôèê O(0, 0) êîîðäèíàòûí ýõýä
òýãø õýìòýé

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
5. “åò ôóíêö:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
5. “åò ôóíêö: Òîîí øóëóóí R = (−∞, +∞) äýýð
òîäîðõîéëîãäñîí f(x) ôóíêöèéí õóâüä:
f(x + T) ≡ f(x) àäèòãàë áèåëýãäýõýýð T > 0 òîî îðøèí
áàéâàë f(x) íü T ³åòýé ôóíêö, T íü óã ôóíêöèéí ³å
ãýíý.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
5. “åò ôóíêö: Òîîí øóëóóí R = (−∞, +∞) äýýð
òîäîðõîéëîãäñîí f(x) ôóíêöèéí õóâüä:
f(x + T) ≡ f(x) àäèòãàë áèåëýãäýõýýð T > 0 òîî îðøèí
áàéâàë f(x) íü T ³åòýé ôóíêö, T íü óã ôóíêöèéí ³å
ãýíý.
³åò ôóíêöèéí ãðàôèê ∀a ∈ R òîîíû õóâüä
· · · , a − 2T ≤ x ≤ a − T, a − T ≤ x ≤ a,
a ≤ x ≤ a + T, a + T ≤ x ≤ a + 2T, · · · T óðòòàé á³õ
õýð÷èì äýýð ÿã àäèëõàí áàéíà.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Íýã õóâüñàã÷èéí ôóíêöÓðâóó ôóíêö
• Õýðýâ f : X → Y ôóíêöèéí y ∈ Y óòãà á³ðò ç°âõ°í íýã x ∈ X
óòãà õàðãàëçäàã áàéâàë óðâóóòàé ôóíêö ãýíý.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
• Õýðýâ f : X → Y óðâóóòàé ôóíêöèéí õóâüä g : Y → X áàéõ
ôóíêöèéã y − f(x)-èéí óðâóó ôóíêö ãýýä x = f−1
(y) ãýæ
òýìäýãëýíý.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
• Õýðýâ f : X → Y óðâóóòàé ôóíêöèéí õóâüä g : Y → X áàéõ
ôóíêöèéã y − f(x)-èéí óðâóó ôóíêö ãýýä x = f−1
(y) ãýæ
òýìäýãëýíý.
Õàðèëöàí óðâóó ôóíêö³³äèéí ãðàôèê y = x øóëóóíû õóâüä òýãø
õýìòýé.
Çóðàã: Õàðèëöàí óðâóó ôóíêö³³ë

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Æèøýý 1.
Ôóíêö Çààã Ìîíîòîí Òýãø/ñîíäãîé “å

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Æèøýý 1.
Ôóíêö Çààã Ìîíîòîí Òýãø/ñîíäãîé “å
f(x) = x2
0 ≤ x2
- òýãø -
f(x) = x3
- °ñí° ñîíäãîé -
f(x) = ex
0 ≤ ex
°ñí° - -
f(x) = sin x −1 ≤ sin x ≤ 1 - ñîíäãîé 2π
f(x) = x + x2
Çóðàã: Æèøýý 2

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Íýã õóâüñàã÷èéí ôóíêö“íäñýí ýëåìåíòàð ôóíêö³³ä
1. Çýðýãò ôóíêö:
f(x) = xa
, a = const

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
f(x) = xa
, a = const D(f) = (−∞; +∞)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
f(x) = xa
, a = const D(f) = (−∞; +∞)
2. Òðàíñöåíäåíò ôóíêö:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
f(x) = xa
, a = const D(f) = (−∞; +∞)
Èëòãýã÷ ôóíêö:f(x) = ax
, a = const, 0 < a =
1

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
f(x) = xa
, a = const D(f) = (−∞; +∞)
, a = const, 0 < a =
1 D(f) = (−∞; +∞)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
f(x) = xa
, a = const D(f) = (−∞; +∞)
, a = const, 0 < a =
1 D(f) = (−∞; +∞)
Ëîãàðèôì ôóíêö:f(x) = loga x, a = const, 0 < a =
1

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
f(x) = xa
, a = const D(f) = (−∞; +∞)
, a = const, 0 < a =
1 D(f) = (−∞; +∞)
1 D(f) = (0; +∞)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
f(x) = xa
, a = const D(f) = (−∞; +∞)
, a = const, 0 < a =
1 D(f) = (−∞; +∞)
1 D(f) = (0; +∞)
3. Òèãîíîìåòðèéí ôóíêö:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
f(x) = xa
, a = const D(f) = (−∞; +∞)
, a = const, 0 < a =
1 D(f) = (−∞; +∞)
1 D(f) = (0; +∞)
f(x) = sin x, f(x) = cos x D(f) = (−∞; +∞)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
f(x) = xa
, a = const D(f) = (−∞; +∞)
, a = const, 0 < a =
1 D(f) = (−∞; +∞)
1 D(f) = (0; +∞)
f(x) = sin x, f(x) = cos x D(f) = (−∞; +∞)
f(x) = tg x, x = π
2 ± 2kπ, f(x) = ctg x, x = 0 ± 2kπ

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
f(x) = xa
, a = const D(f) = (−∞; +∞)
, a = const, 0 < a =
1 D(f) = (−∞; +∞)
1 D(f) = (0; +∞)
f(x) = sin x, f(x) = cos x D(f) = (−∞; +∞)
f(x) = tg x, x = π
2 ± 2kπ, f(x) = ctg x, x = 0 ± 2kπ
4. Òðèãîíîìåòðèéí óðâóó ôóíêö:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
f(x) = xa
, a = const D(f) = (−∞; +∞)
, a = const, 0 < a =
1 D(f) = (−∞; +∞)
1 D(f) = (0; +∞)
f(x) = sin x, f(x) = cos x D(f) = (−∞; +∞)
f(x) = tg x, x = π
2 ± 2kπ, f(x) = ctg x, x = 0 ± 2kπ
f(x) = arcsin x, f(x) = arccos x D(f) = (−1; +1)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
f(x) = xa
, a = const D(f) = (−∞; +∞)
, a = const, 0 < a =
1 D(f) = (−∞; +∞)
1 D(f) = (0; +∞)
f(x) = sin x, f(x) = cos x D(f) = (−∞; +∞)
f(x) = tg x, x = π
2 ± 2kπ, f(x) = ctg x, x = 0 ± 2kπ
f(x) = arcsin x, f(x) = arccos x D(f) = (−1; +1)
, f(x) = arctg x, f(x) = arcctg x D(f) = (−∞; +∞)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
Íýã õóâüñàã÷èéí ôóíêöÝëåìåíòàð ôóíêö³³ä
1. Á³õýë ðàöèîíàëü ôóíêö: m-çýðãèéí îëîí ãèø³³íòýýð
èëýðõèéëýãäýíý.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
P(x) = a0xm
+ a1xm−1
+ · · · + am−1x + am, m ∈ N, ai ∈ R

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
P(x) = a0xm
+ a1xm−1
+ · · · + am−1x + am, m ∈ N, ai ∈ R
2. Áóòàðõàé ðàöèîíàëü ôóíêö:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
P(x) = a0xm
+ a1xm−1
+ · · · + am−1x + am, m ∈ N, ai ∈ R
2. Áóòàðõàé ðàöèîíàëü ôóíêö:Á³õýë ðàöèîíàëü
ôóíêö³³äèéí íîîãäâîð áóþó áóòàðõàéãààð èëýðõèéëýãäýíý.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
P(x) = a0xm
+ a1xm−1
+ · · · + am−1x + am, m ∈ N, ai ∈ R
R(x) =
P(x)
Q(x)
=
a0xm
+ a1xm−1
+ · · · + am−1x + am
b0xn + b1xn−1 + · · · + bn−1x + bn
3. Èððàöèîíàëü ôóíêö:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
P(x) = a0xm
+ a1xm−1
+ · · · + am−1x + am, m ∈ N, ai ∈ R
R(x) =
P(x)
Q(x)
=
a0xm
+ a1xm−1
+ · · · + am−1x + am
b0xn + b1xn−1 + · · · + bn−1x + bn
3. Èððàöèîíàëü ôóíêö:Á³õýë áà áóòàðõàé çýðýãòýé
ôóíêö³³äýýð èëýðõèéëýãäýíý.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
P(x) = a0xm
+ a1xm−1
+ · · · + am−1x + am, m ∈ N, ai ∈ R
R(x) =
P(x)
Q(x)
=
a0xm
+ a1xm−1
+ · · · + am−1x + am
b0xn + b1xn−1 + · · · + bn−1x + bn
ôóíêö³³äýýð èëýðõèéëýãäýíý.f(x) =
√
x, f(x) =
x +
√
x, f(x) = (x2 + 1)(3x2 − 8) + (x + 5
√
x)3

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
P(x) = a0xm
+ a1xm−1
+ · · · + am−1x + am, m ∈ N, ai ∈ R
R(x) =
P(x)
Q(x)
=
a0xm
+ a1xm−1
+ · · · + am−1x + am
b0xn + b1xn−1 + · · · + bn−1x + bn
√
x, f(x) =
x +
√
x, f(x) = (x2 + 1)(3x2 − 8) + (x + 5
√
x)3
4. Òðàíñöåíäåíò ôóíêö:Ðàöèîíàëü áà èððàöèîíàëü
ôóíêö³³äýýñ áóñàä á³õ ôóíêö³³äýýð

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
“íäñýí
ýëåìåíòàð
ôóíêö³³ä
Ýëåìåíòàð
ôóíêö³³ä
Ôóíêöèéí
õÿçãààð
P(x) = a0xm
+ a1xm−1
+ · · · + am−1x + am, m ∈ N, ai ∈ R
R(x) =
P(x)
Q(x)
=
a0xm
+ a1xm−1
+ · · · + am−1x + am
b0xn + b1xn−1 + · · · + bn−1x + bn
√
x, f(x) =
x +
√
x, f(x) = (x2 + 1)(3x2 − 8) + (x + 5
√
x)3
4. Òðàíñöåíäåíò ôóíêö:Ðàöèîíàëü áà èððàöèîíàëü
ôóíêö³³äýýñ áóñàä á³õ ôóíêö³³äýýð
èëýðõèéëýãäýíý.f(x) = sin x, f(x) = x + sin x, f(x) = x
√
2
,
f(x) = 2x
, f(x) = xx
, f(x) = arcsin xÄ.Áàòò°ð email: batturdn@gmail.com Ìàòåìàòèê 1

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
³åèéí õÿçãààð
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Ôóíêöèéí õÿçãààðÕÿçãààðûí óõàãäàõóóí
1. f(x) ôóíêöèéí x0 öýã äýýðõ, ýñâýë x → x0 (èêñ íü èêñ òýã
ð³³ òýì³³ëýõ ³åèéí) õÿçãààð

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
f(x) ôóíêöèéí x0 öýã äýýðõ áàðóóí-°ð°°ñã°ë õÿçãààð
lim
x → x0
(x > x0)
f(x) = lim
x→x0+0
f(x) = b = f(x0 + 0);

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
f(x) ôóíêöèéí x0 öýã äýýðõ áàðóóí-°ð°°ñã°ë õÿçãààð
lim
x → x0
(x > x0)
f(x) = lim
x→x0+0
f(x) = b = f(x0 + 0);
f(x) ôóíêöèéí x0 öýã äýýð ç³³í-°ð°°ñã°ë õÿçãààð
lim
x → x0
(x < x0)
f(x) = lim
x→x0−0
f(x) = b = f(x0 − 0);

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
2. f(x) ôóíêöèéí ∞ ò°ãñã°ëã³é öýã äýýðõ áóþó x → ∞ ³åèéí
õÿçãààð lim
x→∞
f(x) = c;

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
õÿçãààð lim
x→∞
f(x) = c;
f(x) ôóíêöèéí +∞ öýã äýýðõ áóþó x → +∞ ³åèéí
õÿçãààð lim
x→+∞
f(x) = c ;

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
õÿçãààð lim
x→∞
f(x) = c;
õÿçãààð lim
x→+∞
f(x) = c ;
f(x) ôóíêöèéí −∞ öýã äýýðõ áóþó x → −∞ ³åèéí
õÿçãààð lim
x→−∞
f(x) = c

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
õÿçãààð lim
x→∞
f(x) = c;
õÿçãààð lim
x→+∞
f(x) = c ;
f(x) ôóíêöèéí −∞ öýã äýýðõ áóþó x → −∞ ³åèéí
õÿçãààð lim
x→−∞
f(x) = c
3. f(x) ôóíêöèéí x0 öýã äýýðõ ò°ãñã°ëã³é õÿçãààðóóä
lim
x→x0
f(x) = ∞, lim
x→x0
f(x) = +∞, lim
x→x0
f(x) = −∞

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Ôóíêöèéí õÿçãààðf(x) ôóíêöèéí x → x0 ³åèéí õÿçãààð
[Êîøè]: Õýðýâ äóðûí 0 < ε àâàõàä ò³³íýýñ, õàìààðàõ δ = δ(ε) > 0
òîî îëäîîä |x − x0| < δ í°õöëèéã õàíãàõ x-èéí óòãóóäàä
|f(x) − b| < ε áàéâàë f(x) ôóíêöèéí x = x0 öýã äýýðõ õÿçãààð íü
b òîî áàéõ áà lim
x→x0
f(x) = b ãýæ òýìäýãëýíý.

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
x→x0
Æèøýý 2:
f(x) = c = const : lim
x→x0
f(x) = c

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
x→x0
Æèøýý 2:
x→x0
f(x) = c
f(x) = x øóëóóí : lim
x→x0
f(x) = x0

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
x→x0
Æèøýý 2:
x→x0
f(x) = c
f(x) = x øóëóóí : lim
x→x0
f(x) = x0
R(x) = Pn(x)
Qm(x) , Qm(x0) = 0 :
lim
x→x0
R(x) = lim
x→x0
Pn(x)
Qm(x)
=
Pn(x0)
Qm(x0)
= R(x0)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Ôóíêöèéí õÿçãààðx = x0 öýã äýýðõ ôóíêöèéí õÿçãààðûí ä³ðì³³ä
Õýðýâ lim
x→x0
f(x) = a, lim
x→x0
g(x) = b áîë:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Õýðýâ lim
x→x0
f(x) = a, lim
x→x0
g(x) = b áîë:
∃ lim
x→x0
[f(x) ± g(x)] = a ± b = lim
x→x0
f(x) ± lim
x→x0
g(x);

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Õýðýâ lim
x→x0
f(x) = a, lim
x→x0
g(x) = b áîë:
∃ lim
x→x0
[f(x) ± g(x)] = a ± b = lim
x→x0
f(x) ± lim
x→x0
g(x);
∀α = const, ∃ lim
x→x0
[α · f(x)] = α · a = α lim
x→x0
f(x);

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Õýðýâ lim
x→x0
f(x) = a, lim
x→x0
g(x) = b áîë:
∃ lim
x→x0
[f(x) ± g(x)] = a ± b = lim
x→x0
f(x) ± lim
x→x0
g(x);
x→x0
[α · f(x)] = α · a = α lim
x→x0
f(x);
∃ lim
x→x0
[f(x) · g(x)] = a · b = [ lim
x→x0
f(x)] · [ lim
x→x0
g(x)];

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Õýðýâ lim
x→x0
f(x) = a, lim
x→x0
g(x) = b áîë:
∃ lim
x→x0
[f(x) ± g(x)] = a ± b = lim
x→x0
f(x) ± lim
x→x0
g(x);
x→x0
[α · f(x)] = α · a = α lim
x→x0
f(x);
∃ lim
x→x0
[f(x) · g(x)] = a · b = [ lim
x→x0
f(x)] · [ lim
x→x0
g(x)];
Õýðýâ g(x) = 0 áà b = 0 áîë:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Õýðýâ lim
x→x0
f(x) = a, lim
x→x0
g(x) = b áîë:
∃ lim
x→x0
[f(x) ± g(x)] = a ± b = lim
x→x0
f(x) ± lim
x→x0
g(x);
x→x0
[α · f(x)] = α · a = α lim
x→x0
f(x);
∃ lim
x→x0
[f(x) · g(x)] = a · b = [ lim
x→x0
f(x)] · [ lim
x→x0
g(x)];
Õýðýâ g(x) = 0 áà b = 0 áîë:∃ lim
x→x0
f(x)
g(x) = a
b =
lim
x→x0
f(x)
lim
x→x0
g(x) ;

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Õýðýâ lim
x→x0
f(x) = a, lim
x→x0
g(x) = b áîë:
∃ lim
x→x0
[f(x) ± g(x)] = a ± b = lim
x→x0
f(x) ± lim
x→x0
g(x);
x→x0
[α · f(x)] = α · a = α lim
x→x0
f(x);
∃ lim
x→x0
[f(x) · g(x)] = a · b = [ lim
x→x0
f(x)] · [ lim
x→x0
g(x)];
x→x0
f(x)
g(x) = a
b =
lim
x→x0
f(x)
lim
x→x0
g(x) ;
Õýðýâ lim
x→x0
f(x) = a, lim
x→x0
g(x) = b áà f(x) ≤ g(x)
áîë:

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Õýðýâ lim
x→x0
f(x) = a, lim
x→x0
g(x) = b áîë:
∃ lim
x→x0
[f(x) ± g(x)] = a ± b = lim
x→x0
f(x) ± lim
x→x0
g(x);
x→x0
[α · f(x)] = α · a = α lim
x→x0
f(x);
∃ lim
x→x0
[f(x) · g(x)] = a · b = [ lim
x→x0
f(x)] · [ lim
x→x0
g(x)];
x→x0
f(x)
g(x) = a
b =
lim
x→x0
f(x)
lim
x→x0
g(x) ;
Õýðýâ lim
x→x0
f(x) = a, lim
x→x0
g(x) = b áà f(x) ≤ g(x)
áîë: lim
x→x0
f(x) ≤ lim
x→x0
g(x);

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Õýðýâ lim
x→x0
f(x) = a, lim
x→x0
g(x) = b áîë:
∃ lim
x→x0
[f(x) ± g(x)] = a ± b = lim
x→x0
f(x) ± lim
x→x0
g(x);
x→x0
[α · f(x)] = α · a = α lim
x→x0
f(x);
∃ lim
x→x0
[f(x) · g(x)] = a · b = [ lim
x→x0
f(x)] · [ lim
x→x0
g(x)];
x→x0
f(x)
g(x) = a
b =
lim
x→x0
f(x)
lim
x→x0
g(x) ;
Õýðýâ lim
x→x0
f(x) = a, lim
x→x0
g(x) = b áà f(x) ≤ g(x)
áîë: lim
x→x0
f(x) ≤ lim
x→x0
g(x);
Õýðýâ x = x0 ³åä f(x) = g(x) áà ∃ lim
x→x0
g(x) = b áàéâàë
∃ lim
x→x0
f(x) = lim
x→x0
g(x) = b

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Õýðýâ f(x), φ(x), g(x) ôóíêö³³ä X îëîíëîã äýýð
òîäîðõîéëîãäñîí, lim
x→x0
f(x) = lim
x→x0
g(x) = b á°ã°°ä
f(x) ≤ φ(x) ≤ g(x)
òýíöýòãýë áèø (ÿäàæ x = x0 ³åä) áèåëýãäýæ áàéâàë
∃ lim
x→x0
φ(x) = b;

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
x→x0
f(x) = lim
x→x0
f(x) ≤ φ(x) ≤ g(x)
∃ lim
x→x0
φ(x) = b;
y = f[φ(x)] °ã°ãäñ°í ãýæ ³çüå. Õýðýâ ∃ lim
x→x0
φ(x) = u0
áà lim
u→u0
f(u) = f(u0) áèåëýãäýæ áàéâàë:
∃ lim
x→x0
f[φ(x)] = lim
u→u0
f(u) = f(u0)

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Ôóíêöèéí õÿçãààðx → ∞, x → +∞, x → −∞ ³åèéí õÿçãààð
x→x0
f(x) = lim
x→x0
f(x) ≤ φ(x) ≤ g(x)
∃ lim
x→x0
φ(x) = b;

Íýã
ôóíêö,
ò³³íèéã
àðãóóä
Ôóíêöèéí
õÿçãààð
f(x)
ôóíêöèéí
x → x0
x →
∞, x →
+∞, x →
−∞ ³åèéí
õÿçãààð
Ôóíêöèéí õÿçãààðx → ∞, x → +∞, x → −∞ ³åèéí õÿçãààð
x→x0
f(x) = lim
x→x0
f(x) ≤ φ(x) ≤ g(x)
∃ lim
x→x0
φ(x) = b;
y = f[φ(x)] °ã°ãäñ°í ãýæ ³çüå. Õýðýâ ∃ lim
x→x0
φ(x) = u0
áà lim
u→u0
f(u) = f(u0) áèåëýãäýæ áàéâàë:
∃ lim
x→x0
f[φ(x)] = lim
u→u0
f(u) = f(u0)

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