Zestaw nr 3: Granica i pochodna funkcji
Zadanie 1. Oblicz granice funkcji:
"
" "
2x2 + 1 x2 + 3
x+2+ x-5
(1) lim (2) lim (3) lim
x
2
x" x" - 9
x"
x"+ 2x - 1 7x
" " "
7x+2
2x+5
(4) lim x + 2 - x - 5 (5) lim x2 - 3 - 2x2 + 2 (6) lim
2x
x" x-" x-"
7x+2
x-3 1+ex 2-e-x
(7) lim (8) lim (9) lim
x 2x-6 x2+6
x" x-" x"
1+x
1 1
2x-6
x x
(10) lim e (11) lim xe (12) lim e ln x2
x-" x" x-"
sin(x2+3x)
7x3+2x2-7 4x2-5x
(13) lim (14) lim (15) lim
x2+1 2x4-1 3x10+13
x-" x" x-"
" " "
x x
x 1
(16) lim x2 + x (17) lim 2x + 3x (18) lim ( 4 - ln(1 + ))
x
x" x" x"
Zadanie 2. Oblicz granice funkcji:
4 -2 x2+2x
(1) lim (2) lim (3) lim
ln(x+1) 2x2+x
x0+ x0
x0- ln(x+1)
x2-3x+2 x2+4 x2-4
(4) lim (5) lim (6) lim
x-1 x-2
x1 x2
x2+ x-2
1+x
1 -5
x2-4x+3
(7) lim ln (8) lim ln (9) lim e
x-1+ x+1 x1- x2-1 x3+
1+x
1 1
x2-4x+3
(10) lim e (11) lim (12) lim
x3- x1+ (x-1)2 x1- x-1
"
e" x+1 1 7x3+2x2-7
"
(13) lim (14) lim (15) lim
x
e x-1 x2+1
x" x1 x-"
Zadanie 3. Oblicz granice funkcji wykorzystując regułę de l Hospitala :
1
x tg(x-1)
4x e
(1) lim (2) lim (3) lim
sin 3x
x0
x0- ln(x+1) x1 4x-4
1
x5-32 ex-e-x
x
(4) lim (5) lim xe (6) lim
x-2 sin x
x2 x0
x0+
2x2-5 2-2x
(7) lim (8) lim (9) lim x2 ln x
ex ln x
x" x1
x0+
1-cos2 x ln x e-x
(10) lim (11) lim (12) lim
sin x x2+1 x3+1
x0 x" x-"
Zadanie 4. Oblicz pochodne funkcji:
7
"
(1) f(x) = 2x3 + 4x2 - 5x + 4 (2) f(x) = (3) f(x) = x2ex
4
x+3
2x - 1 x2 - 1
(4) f(x) = " (5) f(x) = " (6) f(x) = 4 cos(3x) - 2 sin2 x
x
2
e3x 4x 4 3 5
(7) f(x) = (8) f(x) = (9) f(x) = - +
x2+4 5+2x x3 x2 x
(10) f(x) = e7+3 ln x (11) f(x) = sin 4x2 + 11 (12) f(x) = e2x cos x
1-2x 5
(13) f(x) = (14) f(x) = ln (15) f(x) = ln (sin 3x)
1-x x-2
ex
(16) f(x) = x3 ln(5x) (17) f(x) = ln(x2 - 1) (18) f(x) = ln
sin x
(19) f(x) = x3 ln2(x) (20) f(x) = ex ln2(x) (21) f(x) = sin2 x + cos2 x
Katedra Ekonometrii UA

1
Odpowiedzi
Zadanie 1.
1
(1) 2 (2) (3) 0
7
35
2
(4) 0 (5) -" (6) e
(7) e-21 (8) 0 (9) 0
1
2
(10) e (11) " (12) "
(13) -" (14) 0 (15) 0
(10) 1 (17) 3 (18) 1
Zadanie 2.
(1) " (2) " (3) 2
(4) -1 (5) " (6) 4
(7) " (8) " (9) "
(10) 0 (11) " (12) -"
(13) 1 (14) nie istnieje (15) -"
Zadanie 3.
4 1
(1) (2) 0 (3)
3 4
(4) 80 (5) " (6) 2
(7) 0 (8) 2 (9) 0
(10) 0 (11) 0 (12) -"
Zadanie 4.
7 x"
(1) f (x) = 8x + 6x2 - 5 (2) f (x) = - (3) f (x) = 2xex + x2ex
4
4
(x+3) x+3
"
"
1
(4) f (x) = 2 (5) f (x) = 2 x - x2 - 1 (6) f (x) = -4 cos x sin x - 12 sin 3x
3
2
2x
3x2-2x+12 e3x
( )
6 5 12
(7) f (x) = (8) f (x) = 20 (2x + 5)-2 (9) f (x) = - -
x3 x2 x4
(x2+4)2
3
(10) f (x) = e3 ln x+7 = 3e7x2 (11) f (x) = 8x cos 4x2 + 11 (12) f (x) = e2x cos x (2 cos x - 2x sin x)
x
1 1 5 1 2 cos 3x
"
(13) f(x) = - (14) f (x) = - x - (15) f(x) = 3
2 5 5 sin 3x
(x-2)2
(x-1)2 1 (1-2x)
1-x
cos x
(16) f (x) = x2 + 3x2 ln 5x (17) f (x) = 2x x (18) f (x) = 1 -
2
-1 sin x
2
(20) f (x) = 2x2 ln x + 3x2 ln2 x (20) f (x) = (ln2 x)ex + (ln x) ex (21) f (x) = 0
x
Katedra Ekonometrii UA

2