Question

\( \lim_{x \to \infty} \frac{3x^3 + 2x^2 - 7x + 9}{4x^3 + 9x - 2} \) is equal to

• A. \( \frac{2}{9} \)
• B. \( \frac{1}{2} \)
• C. \( -\frac{9}{2} \)
• D. \( \frac{3}{4} \)
• E. \( \frac{9}{2} \)

Hint:

Only highest power terms matter for limits at infinity.

Answer 1

The Correct Option is D

Concept: For rational functions: \[ \lim_{x \to \infty} \frac{ax^n + \cdots}{bx^n + \cdots} = \frac{a}{b} \]

Step 1: Identify highest degree terms.
Numerator → \( 3x^3 \)
Denominator → \( 4x^3 \)

Step 2: Divide numerator and denominator by \( x^3 \).
\[ = \frac{3 + \frac{2}{x} - \frac{7}{x^2} + \frac{9}{x^3}}{4 + \frac{9}{x^2} - \frac{2}{x^3}} \]

Step 3: Take limit as \( x \to \infty \).
All terms with \( \frac{1}{x} \to 0 \)

Step 4: Simplify expression.
\[ = \frac{3}{4} \]

Step 5: Final result.
\[ \frac{3}{4} \]