limn→∞ pn+qn2n, p,q>0 equals

$lim_{n \to \infty} {(\frac{\sqrt[n]{p} + \sqrt[n]{q}}{2})}^{n}, p, q > 0$ equals

A
1
B
$\sqrt{p q}$
C
$p q$
D
$\frac{p q}{2}$

Solution:

$lim_{n \to \infty} {(\frac{\sqrt[n]{p} + \sqrt[n]{q}}{2})}^{n}$

$= lim_{x \to 0} {(\frac{p^{x} + q^{x}}{2})}^{1 / x},$ where $n = \frac{1}{x}$

$= lim_{x \to 0} {[1 + \frac{1}{2} \{(p^{x} - 1) + (q^{x} - 1)\}]}^{1 / x} = \sqrt{p q}$

Related content

JEE Main 2023 Question Papers with Solutions

JEE Main 2024 Syllabus

Best Books for JEE Main 2024

JEE Advanced 2024: Exam date, Syllabus, Eligibility Criteria

JEE Main 2024: Exam dates, Syllabus, Eligibility Criteria

JEE 2024: Exam Date, Syllabus, Eligibility Criteria

NCERT Solutions For Class 6 Maths Data Handling Exercise 9.3

JEE Crash Course – JEE Crash Course 2023

NEET Crash Course – NEET Crash Course 2023

JEE Advanced Crash Course – JEE Advanced Crash Course 2023