Question

Evaluate the limit: \[ \lim\limits_{x \to \infty} \frac{[2x - 3]}{x}. \]

• A. \( 0 \)
• B. \( \infty \)
• C. \( -3 \)
• D. \( 2 \)

Hint:

For limits of the form \( \lim\limits_{x \to \infty} \frac{ax + b}{x} \), divide every term by \( x \) and simplify using the fact that \( \frac{b}{x} \to 0 \) as \( x \to \infty \).

Answer 1

The Correct Option is D

Step 1: Divide by \( x \) in the Numerator and Denominator \[ \lim\limits_{x \to \infty} \frac{2x - 3}{x}. \] Divide each term in the numerator by \( x \): \[ \lim\limits_{x \to \infty} \left(\frac{2x}{x} - \frac{3}{x} \right). \] \[ = \lim\limits_{x \to \infty} \left( 2 - \frac{3}{x} \right). \] Step 2: Apply the Limit Since \( \frac{3}{x} \to 0 \) as \( x \to \infty \), we get: \[ 2 - 0 = 2. \] Step 3: Conclusion
Thus, the correct answer is: \[ \mathbf{2}. \]