Find the value of (x^2-4)(x^2-x-2)/(x^2+x-6)^2,x-->2

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The limit `lim_(x->2) ((x^2-4)(x^2-x-2))/(x^2+x-6)^2` has to be evaluated.

`lim_(x->2) ((x^2-4)(x^2-x-2))/(x^2+x-6)^2`

=> `lim_(x->2) ((x - 2)(x + 2)(x^2-2x+x-2))/(x^2+3x-2x-6)^2`

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

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

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

substitute x = 2

=> `(4*3)/25`

=> `12/25`

The limit `lim_(x->2) ((x^2-4)(x^2-x-2))/(x^2+x-6)^2 = 12/25`

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