Question

Wyznacz granicę:
a) lim (x^2)/[e^(x^2)]
x→+∞
b) lim (sinx-x)/(x)
x→0+
c) lim (x)/[ln(1+x^2)]
x→-∞
d) lim [1-(x^2/2)-cosx]/(x^3)
x→0+

Answer 1

W zapisie pomijam do czego dąży x, bede pisał tylko lim

korzystamy wszędzie z reguły de l'Hospitala:
a)
lim (x^2)/[e^(x^2)] = [∞/∞] = lim( 2x / (2x * e^(x^2))) =
= lim(1/( e^(x^2))) = 1/∞ = 0

b)
lim (sinx-x)/(x) = [0/0] = lim(cosx - 1) = 0

c)
lim (x)/[ln(1+x^2)] = [-∞/∞] = lim(1/(1/(1+x^2)*2x)) =
= lim((1+x^2)/2x) = lim((x(1/x^2 +1))/2) = -∞

d)
lim [1-(x^2/2)-cosx]/(x^3) = [0/0] = lim((-x+sinx)/3x^2) = [0/0] =
= lim((-1+cosx)/6x) = [0/0] = lim(-sinx/6) = 0