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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 harmonic 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 +C }{3}


Option: 2

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


Option: 3

\frac{3}{AB+BC+CA}


Option: 4

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


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


HM= \frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+\dots +\frac{1}{x_{n}}}
It follows that the harmonic 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 )
HM= \frac{\left ( n= 3 \right )}{\frac{1}{\lim_{x\rightarrow q}f\left ( x \right )}+\frac{1}{\lim_{x\rightarrow q}g\left ( x \right )}+\frac{1}{\lim_{x\rightarrow q}h\left ( x \right )}}\quad \dots\left ( iv \right )

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

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

 

Posted by

Gautam harsolia

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