Answer
The equation of the tangent line $l$ of the curve at point $A$ is $$(l): y=\frac{1}{3}x+\frac{2}{3}$$
Work Step by Step
$$y=f(x)=\frac{2x+1}{x+2}$$ Given point $A (1,1)$
According to definition, the slope of the tangent line $l$ at the given point $A$ is $$m_l=\lim\limits_{x\to1}\frac{f(x)-f(1)}{x-1}$$$$m_l=\lim\limits_{x\to1}\frac{\frac{2x+1}{x+2} -1}{x-1}$$$$m_l=\lim\limits_{x\to1}\frac{\frac{x-1}{x+2}}{x-1}$$$$m_l=\lim\limits_{x\to1}\frac{x-1}{(x+2)(x-1)}$$$$m_l=\lim\limits_{x\to1}\frac{1}{x+2}$$$$m_l=\frac{1}{1+2}=\frac{1}{3}$$
So, the tangent line $l$ of the given function at the given point $A$ would have the form $$(l): y=\frac{1}{3}x+b$$
Since the tangent line $l$ passes through point $A(1,1)$, we have $$\frac{1}{3}\times1+b=1$$$$\frac{1}{3}+b=1$$$$b=\frac{2}{3}$$
Therefore, the equation of the tangent line $l$ of the curve at point $A$ is $$(l): y=\frac{1}{3}x+\frac{2}{3}$$
