Question

The value of \( \lim_{x\to\infty} \frac{4x^3 - x + 1}{x^2 - 4x(1-x^2)} \) is

• A. 0
• B. 1
• C. -1
• D. \(\infty\)

Hint:

For limits at infinity of rational functions:

• If degree of numerator $\lt $ degree of denominator, limit is 0.

• If degree of numerator $\gt $ degree of denominator, limit is \(\pm\infty\).

• If degree of numerator = degree of denominator, limit is the ratio of leading coefficients.
This shortcut can save a lot of time.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
Evaluate lim(x→1) (x³-1)/(x-1).

Step 2: Key Formula (Alternate):
Use standard limit: lim(x→a) (xⁿ-aⁿ)/(x-a) = n·aⁿ⁻¹.

Step 3: Detailed Explanation:
Here n=3, a=1. Result = 3·1² = 3. This avoids factorization or L'Hôpital.

Step 4: Final Answer:
Limit is 3.