Question

. Suppose that ƒ is an odd function of x. Does knowing that limxS0+ ƒ(x) = 3 tell you anything about limxS0- ƒ(x)? Give reasons for your answer.

Answer 1

Answer:

On applying Componendo and Dividendo, we get

\begin{gathered}\rm \: \dfrac{x + y + 2 \sqrt{xy} }{x + y - 2 \sqrt{xy} } = \dfrac{3 + 1}{3 - 1} \\ \end{gathered}

Let assume that x and y are two positive numbers such that x > y and according to statement, sum of these two numbers is 6 times their geometric mean.

So,

\begin{gathered}\rm \: x + y = 6 \sqrt{xy} \\ \end{gathered}

x+y=6

xy

can be further rewritten as

\begin{gathered}\rm \: x + y = 3 \times 2 \sqrt{xy} \\ \end{gathered}

x+y=3×2

xy

\rm \: \dfrac{x + y}{2 \sqrt{xy} } = 3

x+y=3×2

xy

\rm \: \dfrac{x + y}{2 \sqrt{xy} } = 3

2

xy

x+y

=3

can be further rewritten as

\begin{gathered}\rm \: \dfrac{x + y}{2 \sqrt{xy} } = \dfrac{3}{1} \\ \end{gathered}

2

xy

x+y

=

1

3

On applying Componendo and Dividendo, we get

\begin{gathered}\rm \: \dfrac{x + y + 2 \sqrt{xy} }{x + y - 2 \sqrt{xy} } = \dfrac{3 + 1}{3 - 1} \\ \end{gathered}