Question

Evaluate \[ \lim_{x \to \infty} \frac{(3 - x)^{25} (6 + x)^{35}}{(12 + x)^{38} (9 - x)^{22}} = ? \]

• A. \(3^{60}\)
• B. \(-1\)
• C. 1
• D. 0

Hint:

Use dominant term approximation for large \(x\) and simplify powers and signs carefully.

Answer 1

The Correct Option is B

Rewrite terms to dominant powers of \(x\): \[ (3 - x)^{25} \approx (-x)^{25},
(6 + x)^{35} \approx x^{35},
(12 + x)^{38} \approx x^{38},
(9 - x)^{22} \approx (-x)^{22}. \] Simplify: \[ \frac{(-x)^{25} x^{35}}{x^{38} (-x)^{22}} = \frac{(-1)^{25} x^{60}}{(-1)^{22} x^{60}} = \frac{-1}{1} = -1. \]