Question

$\lim _{n \rightarrow \infty}\left(\frac{\sqrt[n]{p}+\sqrt[n]{q}}{2}\right)^{n}\(, p, \)q>0$ is equal to (A) 1 (B) \(\sqrt{\mathrm{pq}}\) (C) pq (D) \(\frac{\mathrm{pq}}{2}\)

Short answer

(A) 1 (B) p (C) q (D) pq Answer: (A) 1

Step-by-step solution

Expand the expression

Expand the given expression as follows: \(\lim_{n \rightarrow \infty}\left(\frac{\sqrt[n]{p}+\sqrt[n]{q}}{2}\right)^{n}=\lim_{n \rightarrow \infty}\left(\frac{p^{\frac{1}{n}}+q^{\frac{1}{n}}}{2}\right)^{n}\).

Apply the limit

Apply the limit to the simplified expression: \(\lim_{n \rightarrow \infty}\left(\frac{p^{\frac{1}{n}}+q^{\frac{1}{n}}}{2}\right)^{n}\).

Evaluate the limit of \(p^{\frac{1}{n}}\) and \(q^{\frac{1}{n}}\) as \(n \rightarrow \infty\)

When n approaches infinity, \(p^{\frac{1}{n}}\) and \(q^{\frac{1}{n}}\) both approach 1: \(\lim_{n \rightarrow \infty}p^{\frac{1}{n}}=1\) and \(\lim_{n \rightarrow \infty}q^{\frac{1}{n}}=1\).

Substitute the limits in the expression

Substitute the limits of \(p^{\frac{1}{n}}\) and \(q^{\frac{1}{n}}\) in the expression (keeping the denominator as is): \(\lim_{n \rightarrow \infty}\left(\frac{1+1}{2}\right)^{n}= \lim_{n \rightarrow \infty}\left(1\right)^{n}\).

Apply the limit to the expression

Finally, apply the limit to the expression: \(\lim_{n \rightarrow \infty}\left(1\right)^{n}=1\). Comparing our result with the given options, the correct answer is: (A) 1