One mistake (and maybe many)! First you assume h to be a small perturbation of x from x_0. Thus you define f as a function of x=x_0+h. But then you set h to be x!!! So the new x(=h) is not same as the variable x in the function y=g(x)=x+k, i.e. you cannot find the intersecting point of g and l by the way you did.
Also you cannot set the new x(=h) to be x_0!!!
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