WBJEE · Maths · Limits
Let for all \(x>0, f(x)=\lim _{n \rightarrow \infty} n\left(x^{1 / n}-1\right),\) then
- A \(f(x)+f\left(\frac{1}{x}\right)=1\)
- B \(f(x y)=f(x)+f(y)\)
- C \(f(x y)=x f(y)+y f(x)\)
- D \(f(x y)=x f(x)+y f(y)\)
Answer & Solution
Correct Answer
(B) \(f(x y)=f(x)+f(y)\)
Step-by-step Solution
Detailed explanation
We have, \(f(x)=\lim _{n \rightarrow \infty} n\left(x^{1 / n}-1\right)\) \(=\lim _{n \rightarrow \infty} \frac{x^{1 / n}-1}{1 / n}\) Let \(\quad \frac{1}{n}=y\) \(\begin{aligned} \therefore \quad f(x)=& \lim _{y \rightarrow 0} \frac{x^{y}-1}{y} \\ &=\log x \end{aligned}\)…
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