Question

Evaluate \(\lim\limits_{x \to \infty} \dfrac{5x^3 - x^2 \sin 5x}{x^3 \cos 4x + 7|x|^3 - 4|x| + 3}\)

• A. \(\dfrac{5}{4}\)
• B. \(-\dfrac{5}{4}\)
• C. \(\dfrac{5}{7}\)
• D. \(-\dfrac{5}{7}\)

Hint:

For limits at infinity, compare highest degree terms in numerator and denominator.

Answer 1

The Correct Option is C

Use asymptotic dominance:
\[ \text{Numerator } \approx 5x^3 \quad \text{(since } -x^2 \sin 5x = O(x^2))
\text{Denominator } \approx 7x^3 \quad \text{(since others are of lower order)}
\Rightarrow \lim = \dfrac{5x^3}{7x^3} = \dfrac{5}{7} \]