Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade None
Question:
Consider the function $f(x) = \left(\frac{ax+1}{bx+2}\right)^x$, where $a, b > 0$ the $\lim_{x \to \infty} f(x)$ is:
exists for all values of $a$ and $b$
zero for $a b$
None
$e^{1/(b-a)}$ or $e^{-1/(b-a)}$, if $a = b$
Step-by-Step Solution
Key Concept: Differential inequalities can be solved by analyzing the sign of expressions and using monotonicity properties.
Given $f'(x) - \sin x \cdot f(x) \le -\sin x$, rearranging gives $f'(x) \le \sin x(f(x) - 1)$. Analyzing this inequality shows that $f(x) \le 1$ for the domain where the inequality holds, implying $f(x) \in (-\infty, 1]$.
Correct Answer: 2,3,4