Question 370647
The left-hand limit for f(x) is of the form 0/0, and so use L'Hopitals Rule:  Get the limit of {{{k(sec(kx))^2/1 }}} as x approaches 0 from the left.  The left hand limit is then equal to k.  The right-hand limit as x approaches 0 is {{{2k^2}}}.
Thus {{{2k^2 = k}}}, (by condition of continuity)
or k = 0 or 1/2.  Discard the value k = 0.  
Then the answer is k = 1/2.