Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
$\lim_{n \to \infty} \left(\frac{\sqrt[n]{p} + \sqrt[n]{q}}{2}\right)^n$, $p, q > 0$ equals :
1
\sqrt{pq}
pq
\frac{pq}{2}
Step-by-Step Solution
Key Concept: Convert the indeterminate form $\infty^0$ to $\infty \cdot 0$ using logarithms, then apply Taylor expansions for $a^{1/n}$ terms.
Find $L = \lim_{n \to \infty} \left(\frac{p^{1/n} + q^{1/n}}{2}\right)^n$. Taking logarithm: $\ln L = \lim_{n \to \infty} n\ln\left(\frac{p^{1/n} + q^{1/n}}{2}\right)$. Using Taylor expansion $p^{1/n} = 1 + \frac{\ln p}{n} + O(1/n^2)$, the limit evaluates to $\frac{\ln p + \ln q}{2} = \ln\sqrt{pq}$, so $L = \sqrt{pq}$.
Correct Answer: 2