1) Lim(3x²+x-1)/(2x³+3) =Lim(3/x+1/x²-1/x³)/(2+3/x³)= 0 .
x→∞ x→∞
-----------------------
2) Lim(x² -x-2)/(3x²+2x-1) * * * неопр типа 0/0 * * *
x→ -1
= Lim(x+1)(x-2)/3(x+1)(x-1/3) =Lim(x-2)/(3x -1) =(-1-2)(-3 -1) = 3 . x→ -1 x→ -1 x→ -1
-----------------------
3)Lim(√(9+x) -√(9-x))/(x²+6x) ;* * * неопр типа 0/0 * * *
x→0
(√(9+x) -√(9-x))/(x²+6x)=(√(9+x) -√(9-x))(√(9+x) +√(9-x)) /(x(x+6)(√(9+x)+√(9-x)) =
((√(9+x))² -(√(9-x))² )/x(6+x)(√(9+x)+√(9-x)) =(9+x -9+x)/x(6+x)(√(9+x)+√(9-x))=
2x/x(6+x)(√(9+x)+√(9-x)) = 2/(6+x)(√(9+x)+√(9-x)) .
Lim(√(9+x) -√(9-x))/(x²+6x) =
x→0
Lim2/(6+x)(√(9+x)+√(9-x))=2/(6+0)(√(9+0) +√(9+0))=1/18.
x→0
-----------------------
4) Lim (sin²x/3)/x² = Lim (sin²x/3)/9*(x/3)² =(1/9)*Lim ((sinx/3)/(x/3))² =1/9
x→0 x→0 x→0
*** z =x/3 ; x →0 ⇒z→0 Lim (sinz/z)² =Lim (sinz/z)*Lim (sinz/z) =1*1=1 * * *
-----------------------
5) Lim((x-3)/x)^(3x-1)
x→∞
((x-3)/x)) ^(3x-1) =(1 -3/x)^((3x-1) = ( (1+1/(-x/3)) ^(-x/3)*(1-3x)*3/x) =
( (1+1/(-x/3)) ^(-x/3)*3(1/x-3)