Question

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

• A. 0
• B. 1
• C. 3
• D. Limit does not exist

Hint:

Recognizing the standard limit form \( \lim_{x\to a} \frac{x^n - a^n}{x - a} = na^{n-1} \) is the fastest way to solve this type of problem. It's a fundamental limit that's worth memorizing for competitive exams.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
Find derivative of y=xˣ.

Step 2: Key Formula (Alternate):
Use logarithmic differentiation: ln y = x ln x, then differentiate implicitly.

Step 3: Detailed Explanation:
(1/y)·dy/dx = ln x + x(1/x) = ln x + 1. dy/dx = y(1+ln x) = xˣ(1+ln x).

Step 4: Final Answer:
Derivative is xˣ(1+log x).