Wydział ILiŚ, Budownictwo i Transport, sem.1
dr Jolanta Dymkowska
Twierdzenie de L Hospitala
Zad.1 Oblicz granice funkcji:
1 x+1 arctg 2x
1.1 lim 1.2 lim 1.3 lim
ex-1 ln x x2+3x
x0 x" x0
e3x-3x-1 x-arctg x
"ln x
1.4 lim 1.5 lim 1.6 lim
sin2 5x x3
x2-1
x0 x0 x"
"
1-x
1-cos x x2-1+ln x e -1
1.7 lim 1.8 lim 1.9 lim
2x2 ex-e
x0 x1
x1- sin(x-1)
cos x - sin x + 1 ln sin 2x ln ln x
1.10 lim 1.11 lim 1.12 lim
sin 2x
Ä„ - cos x x
x"
x x0+ ln sin 3x
2
ex-e-x-2x ex-1-e1-x-2x+2
1.13 lim 1.14 lim
x-sin x x-1-sin(x-1)
x0 x1
1 1 1 1 1 2
1.15 lim - 1.16 lim - 1.17 lim - ctg x
x ex-1 ln x x-1 x2
x0 x1 x0
"
1 1
1.18 lim - 1.19 lim ( x - ln x )
x2 sin2 x
x0 x"
"
1
x
1.20 lim x e 1.21 lim x ln x 1.22 lim tg x · ln x
x0+ x0+ x0+
1
2
Ä„
x2
1.23 lim tg x · e 1.24 lim x2 e-x 1.25 lim x - tg x
2
Ä„ +
x"
x0- x
2
1.26 lim x2 ln x 1.27 lim x ex 1.28 lim x arctg x
x-" x"
x0+
2
x
1
1.29 lim xx 1.30 lim 1 + 1.31 lim (tg x)tg 2x
x0
x0+ x0+ x
1
1
sin x
1 sin x
x x2
1.32 lim 1.33 lim (ln x) 1.34 lim
x
x" x0
x0+ x
1
1
1
x
arctg x (1+x) - e
x2
x
1.35 lim x 1.36 lim 1.37 lim
x x
x" x0 x0