Question

How many tangent lines to the curve y = x/(x+1) pass through the point (1,2)? At which points do these tangent lines touch the curve?

Answer 1

There are two...
(-0.26794919243173,-0.36602540378558)
(-3.7320508756769,1.36602540378460)

Answer 2

Take the derivative of the curve....
let it represent the slope and set into the point-slope form of a line with the x=1 and y=2...then set the equation of the line equal to the equation of the curve and find the points of intersection....and you are done.