Cho hàm số f(x) = x^2 - 3x + 2

20/25

Cho hàm số \(f\left( x \right) = {x^2} - 3x + 2\).

1. \(\mathop {\lim }\limits_{x \to 1} \frac{{f\left( x \right)}}{{x - 1}} = \)	A. \(\frac{1}{3}\).
2. \(\mathop {\lim }\limits_{x \to 1} \frac{{f\left( x \right)}}{{{x^2} - 1}} = \)	B. \( - \frac{1}{3}\).
3. \(\mathop {\lim }\limits_{x \to 1} \frac{{f\left( x \right)}}{{{x^3} - {x^2} + 2x - 2}} = \)	C. \(1\).
4. Để \(\mathop {\lim }\limits_{x \to 1} \frac{{f\left( x \right)}}{{ax + b}} = 2\) thì \(a + 3b = \)	D. \( - 1\).
	E. \(\frac{1}{4}\).
	F. \( - \frac{1}{2}\).

Đáp án: 1 – __ ; 2 – __ ; 3 – __ ; 4 – __

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Giải thích

\(\mathop {\lim }\limits_{x \to 1} \frac{{f\left( x \right)}}{{x - 1}} = \mathop {\lim }\limits_{x \to 1} \frac{{{x^2} - 3x + 2}}{{x - 1}} = \mathop {\lim }\limits_{x \to 1} \frac{{\left( {x - 2} \right)\left( {x - 1} \right)}}{{x - 1}} = \mathop {\lim }\limits_{x \to 1} \left( {x - 2} \right) = - 1\).

\(\mathop {\lim }\limits_{x \to 1} \frac{{f\left( x \right)}}{{{x^2} - 1}} = \mathop {\lim }\limits_{x \to 1} \frac{{\left( {x - 2} \right)\left( {x - 1} \right)}}{{\left( {x - 1} \right)\left( {x + 1} \right)}} = \mathop {\lim }\limits_{x \to 1} \frac{{x - 2}}{{x + 1}} = - \frac{1}{2}\).

\(\mathop {\lim }\limits_{x \to 1} \frac{{f\left( x \right)}}{{{x^3} - {x^2} + 2x - 2}} = \mathop {\lim }\limits_{x \to 1} \frac{{\left( {x - 2} \right)\left( {x - 1} \right)}}{{\left( {x - 1} \right)\left( {{x^2} + 2} \right)}} = \mathop {\lim }\limits_{x \to 1} \frac{{x - 2}}{{{x^2} + 2}} = - \frac{1}{3}\).

Để \(\mathop {\lim }\limits_{x \to 1} \frac{{f\left( x \right)}}{{ax + b}} = 2\) thì \(ax + b\) có nghiệm bằng \(1\) \( \Leftrightarrow a + b = 0 \Leftrightarrow b = - a\).

Khi đó: \(\mathop {\lim }\limits_{x \to 1} \frac{{f\left( x \right)}}{{ax + b}} = \mathop {\lim }\limits_{x \to 1} \frac{{\left( {x - 2} \right)\left( {x - 1} \right)}}{{a\left( {x - 1} \right)}} = \mathop {\lim }\limits_{x \to 1} \frac{{x - 2}}{a} = - \frac{1}{a} = 2 \Leftrightarrow a = - \frac{1}{2} \Rightarrow b = \frac{1}{2}\).

Vậy \(a + 3b = - \frac{1}{2} + 3 \cdot \frac{1}{2} = 1\).

Đáp án: 1 – D; 2 – F; 3 – B; 4 – C.