Answer
$y=-x+4$
The graph of the function and the tangent line is shown below:
Work Step by Step
$y=\dfrac{x}{x-1},$ at $(2,2)$
The slope of the tangent line at the point $P(a,f(a))$ is given by $m=\lim_{h\to0}\dfrac{f(a+h)-f(a)}{h}$
In this case, $a=2$
Find $f(a+h)$ by substituting $x$ by $2+h$ in $f(x)$ and simplifying:
$f(2+h)=\dfrac{2+h}{2+h-1}=\dfrac{2+h}{1+h}$
Since $f(2)$ is the $y$-coordinate of the point given, $f(2)=2$
Substitute the known values into the formula that gives the slope of the tangent line and evaluate:
$m=\lim_{h\to0}\dfrac{\dfrac{2+h}{1+h}-2}{h}=\lim_{h\to0}\dfrac{\dfrac{(2+h)-2(1+h)}{1+h}}{h}=...$
$...=\lim_{h\to0}\dfrac{\dfrac{2+h-2-2h}{1+h}}{h}=\lim_{h\to0}\dfrac{\dfrac{-h}{1+h}}{h}=...$
$...=\lim_{h\to0}\dfrac{-h}{h(1+h)}=\lim_{h\to0}\dfrac{-1}{1+h}=\dfrac{-1}{1+0}=-1$
The point-slope form of the equation of a line is $y-y_{0}=m(x-x_{0})$. Both the slope of the line and a point through which it passes are known. Substitute them into the formula to obtain the equation of the tangent line:
$y-2=(-1)(x-2)$
$y-2=-x+2$
$y=-x+2+2$
$y=-x+4$
