Question

If $ a, b, c $ are distinct real numbers and: $$ \lim_{x \to \infty} \frac{(b - c)x^2 + (c - a)x + (a - b)}{(a - b)x^2 + (b - c)x + (c - a)} = \frac{1}{2} $$ then what is the value of $ a + 2c $?

• A. \( b \)
• B. \( 2b \)
• C. \( 3b \)
• D. \( 4b \)

Hint:

In rational limits, compare leading coefficients of highest-degree terms as \( x \to \infty \).

Answer 1

The Correct Option is C

Let: \[ f(x) = \frac{(b - c)x^2 + (c - a)x + (a - b)}{(a - b)x^2 + (b - c)x + (c - a)} \] As \( x \to \infty \), dominant terms are degree 2: \[ \lim_{x \to \infty} f(x) = \frac{b - c}{a - b} = \frac{1}{2} \quad \text{(1)} \] Cross multiply: \[ 2(b - c) = a - b \Rightarrow 2b - 2c = a - b \Rightarrow a = 3b - 2c \quad \text{(2)} \] We are to find \( a + 2c \): \[ a + 2c = (3b - 2c) + 2c = \boxed{3b} \]