Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f(x)$ is a polynomial function and $(f(x))^2 + (f'(x))^2 = 0$, then find $\lim_{x \to 0}\frac{f(x)}{f'(x)}\left[\frac{f'(x)}{f(x)}\right]$, (where [.] denotes greatest integer function) is_____.
Step-by-Step Solution
Key Concept: A repeated root means both the function and its derivative vanish at that point, making $\frac{f(x)}{f'(x)}$ approach zero as $x$ approaches the root.
Since $(f(a))^2 + (f'(a))^2 = 0$, both $f(a) = 0$ and $f'(a) = 0$, meaning $x = a$ is a repeated root of $f(x)$. The limit simplifies using the fact that $\frac{f(x)}{f'(x)}$ is bounded near $x = a$, yielding $\lim_{x \to a} \frac{f(x)}{f'(x)} = 0$. Therefore the limit equals $1 - 0 = 1$.
Correct Answer: 2