Answer
The equation of the tangent line is $y=-2x+9$.
Work Step by Step
The slope $m$ of the tangent line of the curve $f(x)$ at $A(a,b)$ is also the slope of the curve at that point, which is calculated by $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
$$g(x)=\frac{x}{x-2}\hspace{1cm}A(3,3)$$
1) Find the slope $m$ of the tangent:
$$m=\lim_{h\to0}\frac{g(a+h)-g(a)}{h}$$
Here $a=3$ and $g(a)=b=3$
$$m=\lim_{h\to0}\frac{g(3+h)-3}{h}=\lim_{h\to0}\frac{\frac{3+h}{3+h-2}-3}{h}=\lim_{h\to0}\frac{\frac{h+3}{h+1}-3}{h}$$ $$m=\lim_{h\to0}\frac{\frac{h+3-3h-3}{h+1}}{h}=\lim_{h\to0}\frac{-2h}{h(h+1)}=\lim_{h\to0}\frac{-2}{h+1}$$
$$m=\frac{-2}{0+1}=-2$$
2) Find the equation of the tangent line at $A(3,3)$:
The tangent line would have this form: $$y=-2x+m$$
Substitute $A(3,3)$ here to find $m$:
$$-2\times3+m=3$$ $$-6+m=3$$ $$m=9$$
So the equation of the tangent line is $y=-2x+9$.
