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  • A function f is continuous, If <img alt="f(a) f\left(\frac{1}{a}\right)=f(a)+f\left(\frac{1}{a}\right)" src="https://entrancecorner.oncodecogs.com/gif.latex?f%28a%29%20f%5Cleft%28%5Cfrac%7B1%7D%7Ba

A function f is continuous, If f(a) f\left(\frac{1}{a}\right)=f(a)+f\left(\frac{1}{a}\right) for all x belongs toD_f and  f(1)>0\\   then find the value of \lim _{a \rightarrow 1} f(a).

Option: 1

0


Option: 2

1


Option: 3

2


Option: 4

None of these


Answers (1)

best_answer

Given function is continuous and f(1)>0\\

Now take a=1 in the given expression  f(a) f\left(\frac{1}{a}\right)=f(a)+f\left(\frac{1}{a}\right)

\begin{aligned} & f(1) f\left(\frac{1}{1}\right)=f(1)+f\left(\frac{1}{1}\right) \\ & (f(1))^2=2 f(1) \end{aligned}

By solving the above equation, the value of f(1) can be either 0 or 2

Since  f(1)>0\\  then here the value of the function should be 2 , 

So, f(1)=2

Hence,   \lim _{a \rightarrow 1} f(a)=2

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