Question:
Evaluate
\[
\lim_{x \to \infty} \frac{(3 - x)^{25} (6 + x)^{35}}{(12 + x)^{38} (9 - x)^{22}} = ?
\]
Evaluate
\[
\lim_{x \to \infty} \frac{(3 - x)^{25} (6 + x)^{35}}{(12 + x)^{38} (9 - x)^{22}} = ?
\]
Show Hint
Use dominant term approximation for large \(x\) and simplify powers and signs carefully.
Updated On: Jun 6, 2025
- \(3^{60}\)
- \(-1\)
- 1
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The Correct Option is B
Solution and Explanation
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: \[ \frac{(-x)^{25} x^{35}}{x^{38} (-x)^{22}} = \frac{(-1)^{25} x^{60}}{(-1)^{22} x^{60}} = \frac{-1}{1} = -1. \]
(6 + x)^{35} \approx x^{35},
(12 + x)^{38} \approx x^{38},
(9 - x)^{22} \approx (-x)^{22}. \] Simplify: \[ \frac{(-x)^{25} x^{35}}{x^{38} (-x)^{22}} = \frac{(-1)^{25} x^{60}}{(-1)^{22} x^{60}} = \frac{-1}{1} = -1. \]
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