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To define the variable k, you can write:

sage: var('k')
k

Concerning problem 5, Sage does not answer the right result:

sage: solve(limit(cos(x), x=pi/2, dir='-') == limit(sin(x+k), x=pi/2 , dir='+'), k)                    
[k == 1/2*pi]

Indeed, 1/2*pi is one of the solutions, but the solutions are all numbers of the form 1/2*pi+n*pi (where n is an integer). So, you have to be careful with symbolic computations.