Question:
Evaluate \(\lim\limits_{x \to \infty} \dfrac{5x^3 - x^2 \sin 5x}{x^3 \cos 4x + 7|x|^3 - 4|x| + 3}\)
Evaluate \(\lim\limits_{x \to \infty} \dfrac{5x^3 - x^2 \sin 5x}{x^3 \cos 4x + 7|x|^3 - 4|x| + 3}\)
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For limits at infinity, compare highest degree terms in numerator and denominator.
Updated On: Jun 4, 2025
- \(\dfrac{5}{4}\)
- \(-\dfrac{5}{4}\)
- \(\dfrac{5}{7}\)
- \(-\dfrac{5}{7}\)
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The Correct Option is C
Solution and Explanation
Use asymptotic dominance:
\[ \text{Numerator } \approx 5x^3 \quad \text{(since } -x^2 \sin 5x = O(x^2))
\text{Denominator } \approx 7x^3 \quad \text{(since others are of lower order)}
\Rightarrow \lim = \dfrac{5x^3}{7x^3} = \dfrac{5}{7} \]
\[ \text{Numerator } \approx 5x^3 \quad \text{(since } -x^2 \sin 5x = O(x^2))
\text{Denominator } \approx 7x^3 \quad \text{(since others are of lower order)}
\Rightarrow \lim = \dfrac{5x^3}{7x^3} = \dfrac{5}{7} \]
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