Cho hàm số f ( x ) thỏa lim x → 1 (f ( x ) − 2)/( x − 1) = 3 . Tính lim x → 1( ( f ( x ) )^2 − 3 f ( x ) + 2)/( 2x^2 − 5x + 3)

Đáp án đúng là: \( - 3\)

Giải chi tiết

Ta có ${\text{lim}}_{x\to 1}\frac{f\left(x\right)-2}{x-1}=3\Rightarrow {\text{lim}}_{x\to 1}\left(f\left(x\right)-2\right)=0\Rightarrow {\text{lim}}_{x\to 1}f\left(x\right)=2$
Vậy ${\text{lim}}_{x\to 1}\frac{{\left(f\left(x\right)\right)}^2-3f\left(x\right)+2}{2x^2-5x+3}={\text{lim}}_{x\to 1}\frac{\left(f\left(x\right)-1\right)\left(f\left(x\right)-2\right)}{\left(2x-3\right)\left(x-1\right)}=3{\text{ lim}}_{x\to 1}\frac{\left(f\left(x\right)-1\right)}{\left(2x-3\right)}=-3$.