If lim _(x → ∞) ( √(x² - x + 1) - ax ) = b, then the ordered pair (a, b) is :

Solution

Maths sir · 4 steps

First choose a so the infinite parts cancel, then rationalise to read off b.

1. Fix a first. For large x, √(x² - x + 1) behaves like x, so the expression grows like (1 - a)x. The limit is a finite number b only if this growth vanishes, so a = 1.
2. Rationalise. With a = 1 the limit has the form ∞ - ∞. Multiply and divide by √(x² - x + 1) + x, so the square root leaves the numerator.

   lim(x → ∞) ((x² - x + 1) - x²)/(√(x² - x + 1) + x) = lim(x → ∞) (-1 + 1/x)/(√(1 - 1/x + 1/x²) + 1)

3. Take the limit. On the right, every 1/x term goes to 0. The fraction becomes -1/(1 + 1) = -1/2, so b = -1/2.
4. So. (a, b) = (1, -1/2), which is option (b).

The idea to keep. When a limit at infinity must be finite, first make the leading terms cancel. That fixes the coefficient. Then rationalise the ∞ - ∞ form to get the leftover constant.

Answer from the NTA final answer key. Solution by PrepWiser.