Question

\[ \lim_{x \to \infty} \left(\frac{x^2}{3x-2} - \frac{x}{3}\right) \]

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

Hint:

For limits at infinity, divide numerator and denominator by highest power of \(x\).

Answer 1

Answer: (E)

Concept: Use algebraic simplification for large \(x\).

Step 1: Combine terms
\[ \frac{x^2}{3x-2} - \frac{x}{3} \] Take common denominator:

Step 2: Simplify expression
\[ = \frac{3x^2 - x(3x-2)}{3(3x-2)} \]

Step 3: Expand
\[ = \frac{3x^2 - 3x^2 + 2x}{9x-6} = \frac{2x}{9x-6} \]

Step 4: Divide numerator and denominator by \(x\)
\[ = \frac{2}{9 - \frac{6}{x}} \]

Step 5: Take limit
\[ x \to \infty \Rightarrow \frac{6}{x} \to 0 \] \[ = \frac{2}{9} \]

Step 6: Final Answer
\[ \boxed{\frac{2}{9}} \]