Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a positive increasing function with $\lim_{x \rightarrow \infty} \frac{f(3x)}{f(x)}=1$. Then $\lim_{x \rightarrow \infty} \frac{f(2x)}{f(x)}=$

JEE Main · 2010 · 11 Apr 2010 · Offline · Limits XII

Let

f : R \to R

be a positive increasing function with

lim_{x \to \infty} \frac{f ( 3 x )}{f ( x )} = 1

. Then

lim_{x \to \infty} \frac{f ( 2 x )}{f ( x )} =

Answer: (D)

Because

f

is positive and increasing, for large

x

we have

0 < f (x) < f (2 x) < f (3 x)

, and dividing by

f (x)

gives

1 < \frac{f ( 2 x )}{f ( x )} < \frac{f ( 3 x )}{f ( x )}

. We are told

\frac{f ( 3 x )}{f ( x )} \to 1

, so the middle ratio is squeezed between

1

and a quantity tending to

1

. By the sandwich theorem

\frac{f ( 2 x )}{f ( x )} \to 1

JEE Main · 2010 · 11 Apr 2010 · Offline · Single correct option