QJEEM10Mac68a: JEE Main 2010 Mathematics - Class 12 JEEM10Mac68a Solution: Let be a positive increasing function with . Then

Question

Verified Solution • 2025–26 Session moderate

Let

f:R \to R

be a positive increasing function with

\mathop {\lim }\limits_{x \to \infty } {{f(3x)} \over {f(x)}} = 1

. Then

\mathop {\lim }\limits_{x \to \infty } {{f(2x)} \over {f(x)}} =

Accepted Answer

f(x)

is a positive increasing function.

\therefore

0 &lt; f\left( x \right) &lt; f\left( {2x} \right) &lt; f\left( {3x} \right)

\Rightarrow 0 &lt; 1 &lt; {{f\left( {2x} \right)} \over {f\left( x \right)}} &lt; {{f\left( {3x} \right)} \over {f\left( x \right)}}

\Rightarrow \mathop {\lim }\limits_{x \to \infty } 1 \le \mathop {\lim }\limits_{x \to \infty } {{f\left( {2x} \right)} \over {f\left( x \right)}} \le \mathop {\lim }\limits_{x \to \infty } {{f\left( {3x} \right)} \over {f\left( x \right)}}

By Sandwich Theorem.

\Rightarrow \mathop {\lim }\limits_{x \to \infty } {{f\left( {2x} \right)} \over {f\left( x \right)}} = 1

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