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Find the  \lim_{x\rightarrow a}\left ( AM \right )where AM  is the arithmetic mean of the reciprocals of the positive n real numbers x_{1},x_{2},x_{3},\cdots x_{n}, and \frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\cdots +\frac{1}{x_{n}}= \lim _{x \rightarrow a}f\left ( x \right )= K^{n}.

Option: 1

\left ( \frac{K}{n} \right )^{n}


Option: 2

\frac{n}{K^{n}}


Option: 3

\frac{K^{n}}{n}


Option: 4

\left ( \frac{n}{K} \right )^{n}


Answers (1)

best_answer

Note that the “Constant multiple law for limits” indicates that \lim _{x \rightarrow a}cf\left ( x \right )= c\cdot \lim _{x \rightarrow a}f\left ( x \right )

The harmonic mean of the positive n real numbers x_{1},x_{2},x_{3},\dots x_{n} is 

AM= \frac{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\dots+\frac{1}{x_{n}}}{n}\quad \dots\left ( i \right )

Again, the provided limit is

\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\dots+\frac{1}{x_{n}}= \lim _{p \rightarrow a}f\left ( x \right )= K^{n}\quad \dots\left ( ii \right )

Derive the following limit using the equation (i), (ii) and applying the Constant multiple law for limits in the following way.

\lim _{x \rightarrow a}\left ( AM \right )
= \lim _{x \rightarrow a}\left [ \frac{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\dots+\frac{1}{x_{n}}}{n} \right ]
= \frac{ \lim _{x \rightarrow a}\left [\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\dots+\frac{1}{x_{n}} \right ]}{ \lim _{x \rightarrow a} n}
= \frac{K^{n}}{n}

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sudhir.kumar

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