Lim(tg^2 3x/ sin^2 5x)^1/x 

1) lim(x->-2) (3x^2+5x-2)/(x^2+3x+2) = lim(x->-2) [(x+2)(3x-1)]/[(x+2)(x+1)] = lim(x->-2) (3x-1)/(x+1) =  = (-6-1)/(-2+1) = (-7)/(-1) = 7 2) lim(x->0) tg 3x*ctg 5x = lim(x->0) tg 3x/tg 5x По следствию из 1 Замечательного предела lim(x->0) sin 3x/x = 3, lim(x->0) sin 5x/x = 5, а этот предел lim(x->0) sin 3x/(x*cos 3x) * (x*cos 5x)/sin 5x = lim(x->0) 3/cos 3x*cos 5x/5 = 3/cos 0*cos 0/5 = 3/5 3) lim(x->0) x/[sqrt3(8-x) - sqrt3(8+x)] = lim(x->0) x*[sqrt3(8-x)^2 + sqrt3(8-x)(8+x) + sqrt3(8+x)^2]/[(8-x) - (8+x)] = = lim(x->0) x*[sqrt3(8-x)^2 + sqrt3(8-x)(8+x) + sqrt3(8+x)^2]/(-2x) = = lim(x->0) [sqrt3(8-x)^2 + sqrt3(8-x)(8+x) + sqrt3(8+x)^2]/(-2) = (4+2*2+4)/(-2) = -6 4) lim(x->+oo) ((x-3)/(x+4))^(x-1) = lim(x->+oo) (1 - 7/(x+4))^(x-1) = lim(x->+oo) (1 - 7/(x+4))^(x+4-5) По 2 Замечательному пределу (1 - 7/(x+4))^(x+4) = e^(-7), а этот предел lim(x->+oo) (1 - 7/(x+4))^(x+4-5) = e^(-7)*lim(x->+oo) (1 - 7/(x+4))^(-5) = e^(-7)*1^(-5) = e^(-7)