Answer
The equation of the tangent line is $y=1$.
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 $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
$$y=f(x)=(x-1)^2+1\hspace{1cm}A(1,1)$$
1) Find the slope $m$ of the tangent: $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
Here $a=1$ and $f(a)=b=1$.
$$m=\lim_{h\to0}\frac{\Big[\Big((1+h)-1\Big)^2+1\Big]-1}{h}$$ $$m=\lim_{h\to0}\frac{\Big((1+h)-1\Big)^2}{h}=\lim_{h\to0}\frac{h^2}{h}=\lim_{h\to0}h$$ $$m=0$$
2) Find the equation of the tangent line at $A(1,1)$:
The tangent line would have this form: $$y=0x+m=m$$
Substitute $A(1,1)$ here to find $m$: $$m=1$$
So the equation of the tangent line is $y=1$.
