Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
The value of $\lambda$ for which $2\left(\text{Lim}_{x \to 0} f(x^3-x^2)\right) = \lambda \left(\text{Lim}_{x \to 0} f(2x^4-x^3)\right)$ is:
Step-by-Step Solution
Key Concept: Analyze the sign of expressions near the limit point by factoring and determining the dominant term's behavior.
For $x \to 0^+$: $x^3 - x^2 = x^2(x-1) \to 0^-$ (negative), and $2x^4 - x^5 = x^4(2-x) \to 0^+$ (positive). Since $\lambda(3) = \lambda(2)$, we have $\lambda = 3$.
Correct Answer: 3