Question

The value of \( \lim_{x \to 0} \frac{ax^3 + bx^2 + cx}{3x^2} \), where \( a, b, c > 0 \), is:

• A. \( (abc)^3 \)
• B. \( abc \)
• C. \( (abc)^{1/3} \)
• D. None of these

Hint:

When evaluating limits involving indeterminate forms, be careful of division by zero. Factor out common terms to simplify.

Answer 1

The Correct Option is D

We are given the limit: \[ L = \lim_{x \to 0} \frac{ax^3 + bx^2 + cx}{3x^2} \] Step 1: Simplify the expression Factor out \( x \) from the numerator: \[ L = \lim_{x \to 0} \frac{x(ax^2 + bx + c)}{3x^2} \] Simplifying: \[ L = \lim_{x \to 0} \frac{ax^2 + bx + c}{3x} \] Step 2: Evaluate the limit As \( x \to 0 \), the expression simplifies to: \[ L = \frac{c}{0} \] Since \( c > 0 \), the limit does not exist, so the correct answer is (d) None of these.