Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f$ be a function given by $f(x) = \begin{cases} \frac{1}{x^{n/2}} \cdot \frac{1}{2^x - 1}, & x \neq 0 \\ \frac{1}{2}, & x = 0 \end{cases}$. Then:
$f$ is continuous on $\mathbb{R}$
$f$ is differentiable on $\mathbb{R}$ and $f'(0)$ equals $-\frac{\ln 2}{12}$
$f$ is not differentiable at $x = 0$
$f$ is differentiable on $\mathbb{R}$ and $f'(0)$ equals $-\frac{\ln 2}{6}$
Step-by-Step Solution
Key Concept: Apply Taylor series expansion of exponential functions to resolve indeterminate forms.
Use the binomial expansion of $2^x = e^{x\ln 2} = 1 + x\ln 2 + \frac{(x\ln 2)^2}{2} + O(x^3)$ to evaluate the limit. (Complete expansion method shown in reference.)
Correct Answer: 1,2