А) 3x³-6x+2 x⁴(3/x-6/x³+2/x⁴) (3/x-6/x³+2/x⁴)
lim -------------- =lim --------------------- =lim ------------------------- =0/8=0
x->∞ 7x+8x⁴ x->∞ x⁴(7/x³+8) x->∞ (7/x³+8)
√(6x+4)-4 0 [ (√(6x+4)-4)(√(6x+4)+4) ]·(√(x+2)+2)
б) lim ----------------=[----] = lim --------------------------------------------------- =
x->2 √(x+2)-2 0 x->2 (√(6x+4)+4) ]·(√(x+2)+2) ( √(x+2)-2 )
[(6x+4)-16] (√(x+2)+2) [(6(x-2)] (√(x+2)+2) 6(√(x+2)+2)
= lim ---------------------------------- = lim --------------------------= lim -------------
x->2 (√(6x+4)+4) ]·((x+2)-4) x->2 (√(6x+4)+4) ]·(x-2) x->2 √(6x+4)+4)
6·(√4+2) 24
= ----------= ------ = 3
√16+4 8
x²+x x(x+1) x+1
в) lim ------------ = lim ----------- =lim ------- = 1/2
x->0 arctg2x x->0 2x x->0 2
2x 2x-6 +6 6
г) lim ( -------)^(7x/3)=lim (-------------)^(7x/3)=lim (1 +---------)^(7x/3)=
x->∞ 2x-6 x->∞ 2x-6 x->∞ 2x-6
6
=lim { (1 +---------)^((2x-6)/6) ) }^ {((6/(2x-6) ) ·(7x/3)} =e^(lim(2·7x/(x-3) )
x->∞ 2x-6 x->∞
=e^ 14