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  • There exist three limits <img alt="\frac{1}{\lim _{x \rightarrow q} f(x)}=A, \frac{1}{\lim _{x \rightarrow q} g(x)}=B, \frac{1}{\lim _{x \rightarrow q} h(x)}=C ; \quad[A, B, C \neq 0]" src="https:/

There exist three limits \frac{1}{\lim _{x \rightarrow q} f(x)}=A, \frac{1}{\lim _{x \rightarrow q} g(x)}=B, \frac{1}{\lim _{x \rightarrow q} h(x)}=C ; \quad[A, B, C \neq 0]  find the arithmetic mean of the limits \lim _{x \rightarrow q} f(x), \lim _{x \rightarrow q} g(x), \lim _{x \rightarrow q} h(x).

Option: 1

\frac{A B+B C+C A}{3}


Option: 2

\frac{A B+B C+C A}{3 A B C}


Option: 3

\frac{3 A B C}{A B+B C+C A}


Option: 4

\frac{ A B C}{3\left ( A B+B C+C A \right )}


Answers (1)

best_answer

The given three limits are
\frac{1}{\lim_{x\rightarrow q}f\left ( x \right )}= A \quad \left [ A\neq 0 \right ]\quad\cdots \left ( i \right )
\frac{1}{\lim_{x\rightarrow q}g\left ( x \right )}= B \quad \left [ B\neq 0 \right ]\quad\cdots \left ( ii \right )
\frac{1}{\lim_{x\rightarrow q}h\left ( x \right )}= C \quad \left [ C\neq 0 \right ]\quad\cdots \left ( iii \right )

Note the following important points.
 

  • The “Sum law for limits” states that \lim_{x\rightarrow a}f\left ( x \right )+\lim_{x\rightarrow a}g\left ( x \right )= \lim_{x\rightarrow a}\left [ f\left ( x \right )+g\left ( x \right )\right ].
  • The arithmetic mean of the positive n real numbers x_{1},x_{2},x_{3},\dots x_{n}   is denoted as AM= \frac{x_{1}+x_{2}+x_{3}+\dots x_{n}}{n}

It follows that the arithmetic mean of the three limits \lim_{x\rightarrow q}f\left ( x \right ),\lim_{x\rightarrow q}g\left ( x \right ),\lim_{x\rightarrow q}h\left ( x \right ) is 
AM=\frac{ \lim_{x\rightarrow q}f\left ( x \right )+\lim_{x\rightarrow q}g\left ( x \right ),+\lim_{x\rightarrow q}h\left ( x \right )}{\left ( n= 3 \right )} \quad \dots\left ( iv \right )
 

Using the equations (i), (ii) and (iii), derive the following from the equation (iv).

AM
= \frac{\frac{1}{A}+\frac{1}{B}+\frac{1}{C}}{3}
= \frac{\frac{BC+CA+AB}{ABC}}{3}
=\frac{AB+BC+CA}{3ABC}
 

Posted by

Gaurav

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