lim x^2-4x+3/x^3-9x x--->-3- Solution lim x->3 ( x-3)/(x^2-4x+3)=(3-3)/(9-12+3)=

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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

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