Official solution
\[f'(x) = 7 - {3 \over 4}{{f\left( x \right)} \over x}\,\,\,\left( {x > 0} \right)\]
Given f(1) \[ \ne \] 4 \[\mathop {\lim }\limits_{x \to {0^ + }} \,xf\left( {{1 \over x}} \right)\, = ?\]
\[{{dy} \over {dx}} + {3 \over 4}{y \over x} = 7\] (This is LDE)
IF \[ = {e^{\int {{3 \over {4x}}dx} }} = {e^{{3 \over 4}\ln \left| x \right|}} = {x^{{3 \over 4}}}\]
\[y.{x^{{3 \over 4}}} = \int {7.{x^{{3 \over 4}}}} dx\]
\[y.{x^{{3 \over 4}}} = 7.{{{x^{{7 \over 4}}}} \over {{7 \over 4}}} + C\]
\[f(x) = 4x + C.{x^{ - {3 \over 4}}}\]
\[f\left( {{1 \over 4}} \right) = {4 \over x} + C.{x^{{3 \over 4}}}\]
\[\mathop {\lim }\limits_{x \to {0^ + }} xf\left( {{1 \over x}} \right) = \mathop {\lim }\limits_{x \to {0^ + }} \left( {4 + C.{x^{{7 \over 4}}}} \right) = 4\]