It's Sufficiently Nested!

$\begin{aligned} \cfrac1{1 + \cfrac1{1 + \cfrac1{ 1+\cfrac1x}}} =\frac34 & \Rightarrow \cfrac{1}{1+\cfrac{1}{1+\cfrac{x}{x+1}} }=\frac{3}{4} \\ \cfrac{1}{1+\cfrac{x+1}{2x+1}}=\frac{3}{4} & \Rightarrow \frac{2x+1}{3x+2} =\frac{3}{4} \\ 8x+4 & = 9x+6 \\ x& =\boxed{-2} \end{aligned}$

$\huge \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{x}}}} = \frac{3}{4}$

$\huge \frac{1}{1 + \frac{1}{1 + \frac{1}{\frac{x + 1}{x}}}} = \frac{3}{4}$

$\huge \frac{1}{1 + \frac{1}{1 + \frac{x}{x + 1}}} = \frac{3}{4}$

$\huge \frac{1}{1 + \frac{1}{\frac{2x + 1}{x + 1}}} = \frac{3}{4}$

$\huge \frac{1}{1 + \frac{x + 1}{2x + 1}} = \frac{3}{4}$

$\huge \frac{1}{\frac{3x + 2}{2x + 1}} = \frac{3}{4}$

$\huge \frac{2x + 1}{3x + 2} = \frac{3}{4}$

$\huge 4(2x + 1) = 3(3x + 2)$

$\huge 8x + 4 = 9x + 6$

$\huge 4 = x + 6$

$\huge -2 = x$

$\huge \boxed{x = -2}$

Take the reciprocal on both sides , then subtract 1 from both sides . Then repeat until you obtain the value of $x$ .

By computing rational equation:

$\begin{aligned} \cfrac1{1 + \cfrac1{1 + \cfrac1{ 1+\cfrac1x}}} =\cfrac34 & \Rightarrow \cfrac{2x+1}{3x+2} = \cfrac{3}{4} \\& \Rightarrow 4(2x+1) = 3(3x+2) \quad\quad\quad\quad\quad{\text{Get the LCD both sides}} \\& \Rightarrow 8x+4 = 9x+6 \\&\Rightarrow 8x = 9x+2 \\&\Rightarrow 8x-9x=2 \\& \Rightarrow -x =2 \end{aligned}$

$\therefore x= -2 \space \square$ .

$\LARGE \text{FIN!!!}$