Let $f$ be a differentiable function such that $f^{\prime}(x)=7-\frac{3}{4} \frac{f(x)}{x}$,
$(x>0)$ and $f(1) \neq 4$. Then $\lim _{x \rightarrow 0^{+}} x f\left(\frac{1}{x}\right):$
Correct Option: , 2
Let $y \quad f(x)$
$\frac{d y}{d x}+\left(\frac{3}{4 x}\right) y=7$
I.F. $=e^{\int \frac{3}{4 x} d x}=e^{\frac{3}{4} \ln x}=x^{\left(\frac{3}{4}\right)}$
Solution of differential equation
$y \cdot x^{\frac{3}{4}}=\int 7 \cdot x^{\frac{3}{4}} d x+C$
$y \cdot x^{\frac{3}{4}}=7 \cdot \frac{x^{\frac{7}{4}}}{\left(\frac{7}{4}\right)}+C=4 x^{\frac{7}{4}}+C$
$y=4 x+C x^{-\frac{3}{4}}$
$\Rightarrow f\left(\frac{1}{x}\right)=\frac{4}{x}+C x^{\frac{3}{4}}$
$\Rightarrow \quad \lim _{x \rightarrow 0^{+}} x \cdot f\left(\frac{1}{x}\right)=\lim _{x \rightarrow 0^{+}}\left(4+C x^{\frac{7}{4}}\right)=4$