`(x(2^x-3^x))/((x^2+x+1)(x-1))>0`

Solve the following rational in equalities (i) ((x-1)(x+2))/((x-3)(x+3)) lt 0 (ii) ((1-x)^(3)(x+2)^(4))/((x+9)^(2)(x-8))ge0 (iii) ((x^(2)-3x+1)^(3))/((x-1)(x+2))le0 (iv) (x(2^(x)-3^(x)))/((x^(2)+x+1)(x-1))gt0 (v) ((x-1)(x-2)(x-3))/((x+1)(x+2)(x+3)) le1 (vi) 1 lt (3x^(2)-7x+8)/(x^(2)+1) le2

lim_(x->0) (x^2-3x+1)/(x-1)

Solve : (i)" "((x-1)\(x-2)(x-3))/((x+1)(x+2)(x+3))" "(ii) " "(x^(4)+x^(2)+1)/(x^(2)+4x-5)lt0

Solve : (i)" "((x-1)\(x-2)(x-3))/((x+1)(x+2)(x+3))" "(ii) " "(x^(4)+x^(2)+1)/(x^(2)+4x-5)lt0

Solve |((x+1),0,0)) , ((2x+1),(x-1),0)) , ((3x+1),(2x-1),(x-2))| =0

I : If (x^(2)+x+1)/(x^(2)+2x+1)=A+B/(x+1)+C/(x+1)^(2) " then "A+B+C=0 . II : If (x^(2)+2x+3)/x^(3)=A/x+B/x(2)+C/x^(3) " then "A+B-C=0 .

If x^(4) - 3x^(2) - 1 = 0 , then the value of (x^(6)-3x^(2)+(3)/(x^(2))-(1)/(x^(6))+1) is :

lim_(x rarr0)((5x^(2)+1)/(3x^(2)+1))^(1/x^(2))

If sqrt((2x^(2)+x+2)/(x^(2)+3x+1))+2"."sqrt((x^(2)+3x+1)/(2x^(2)+x+2))-3=0 , find x.