lim x24x3x39x x3Solution lim x3 x3x24x333912300 It is simpl

Solution

lim x->3 ( x-3)/(x^2-4x+3)=(3-3)/(9-12+3)=0/0 It is simple to notice that if \"3\" cancel the expression from the denominator, that means that 3 is a root of the denominator. We\'ll find the other root, using Viete\'s relationships and knowing that: x1 + x2= -(-4/1) Bt x1=3, as we\'ve noticed earlier, so: 3+x2=4 x2=4-3 x2=1 Knowing the both roots, now we can write the denominator as follows: (x^2-4x+3)=(x-x1)(x-x2) (x^2-4x+3)=(x-3)(x-1) We\'ll put back this late expression into the limit: lim x->3 ( x-3)/(x^2-4x+3)= lim x->3 ( x-3)/(x-3)(x-1) It is obvious that we\'ll simplify the common factor (x-3): lim x->3 ( x-3)/(x-3)(x-1)=lim x->3 1/(x-1) Now we\'ll substitute again with the value \"3\". lim x->3 1/(x-1)=1/(3-1)=1/2