Continuity problem

Question

f(x)= (x^2+4)/(x-2) if x<=3 or kx-3 if x>3

 

Determine a value for k that makes the function continuous on the interval (2,10) for all real values of x in that interval.

Please solve this!

Answer 1

Note that the function is continuous everywhere in (2,10) except possibly at x=3.  In order for f to be continuous at x=3, we need the limit f(x) (as x approaches 3 from the left) to be equal to the limit f(x) (as x approaches 3 from the right).  Then find the limit f(x) (as x approaches 3 from the left) and limit f(x) (as x approaches 3 from the right).  Once you find those, set the left hand limit equal to the right hand limit.  Solve for k.

Answer 2

Since f has to be continuous at x = 3, f(3) = lim(x->3+) f(x). Now f(3) = 13, and lim(x->3+) f(x) = lim(x->3+) (kx - 3) = 3k - 3. So we have 13 = 3k - 3, or 16 = 3k, i.e., 16/3 = k.