Question

Evaluate the limit: \[ \lim_{x \to \infty} \frac{3x+4\cos^2x}{\sqrt{x^2-5\sin^2x}} \]

• A. \( \frac{3}{5} \)
• B. \( \frac{4}{5} \)
• C. \( 3 \)
• D. \( 1 \)

Hint:

For limits involving infinity, divide by the highest degree of \( x \) to simplify terms effectively.

Answer 1

The Correct Option is C

Dividing numerator and denominator by \( x \): \[ \frac{3 + \frac{4\cos^2x}{x}}{\sqrt{1 - \frac{5\sin^2x}{x^2}}} \] For large \( x \), \( \frac{4\cos^2x}{x} \to 0 \) and \( \frac{5\sin^2x}{x^2} \to 0 \), simplifying: \[ \frac{3 + 0}{\sqrt{1 - 0}} = 3 \] Thus, the correct answer is: \[ 3 \]