Let f(x) be defined for all x 0 and be continuous. Let f(x) satisfy f ( (x/y) ) = f(x) - f(y) for all x ,y and f(e) = 1. Then

Question

Let f(x) be defined for all x > 0 and be continuous. Let f(x) satisfy f ( (x/y) ) = f(x) - f(y) for all x ,y and f(e) = 1. Then (A) f(x) is bounded (B) f ( (1/x) ) → 0 as x → 0  (C) x f(x) → 1 as x → 0 (D) f(x) = ln x

Tardigrade Admin · Accepted Answer

Correct answer is (d) f(x) = ln x

Q. Let f(x) be defined for all x > 0 and be continuous. Let f(x) satisfy $f (\frac{x}{y}) = f (x) - f (y)$ for all $x, y$ and $f (e) = 1$ . Then

f(x) is bounded

$f (\frac{1}{x}) \to 0$ as $x \to 0$

$x f (x) \to 1$ as $x \to 0$

f(x) = ln x

Correct answer is (d) f(x) = ln x

Q. Let f(x) be defined for all x > 0 and be continuous. Let f(x) satisfy f(yx​)=f(x)−f(y) for all x,y and f(e)=1. Then

f(x) is bounded

f(x1​)→0 as x→0

xf(x)→1 as x→0

f(x) = ln x

Correct answer is (d) f(x) = ln x

Q. Let f(x) be defined for all x > 0 and be continuous. Let f(x) satisfy $f (\frac{x}{y}) = f (x) - f (y)$ for all $x, y$ and $f (e) = 1$ . Then

$f (\frac{1}{x}) \to 0$ as $x \to 0$

$x f (x) \to 1$ as $x \to 0$