If f(x)= array ll 1-x^m 1-x & if x 1 2 m-1 & if x=1 array . and the function is discontinuous at x=1 , then

Step-by-step solution

A function f(x) is continuous at x=a if _ x a f(x) = f(a) . Given f(x) = 1-x^m 1-x for x 1 , we calculate the limit as x 1 : _ x 1 f(x) = _ x 1 1-x^m 1-x . Using L'Hopital's rule or the standard limit formula _ x 1 x^m - 1 x - 1 = m , we get: _ x 1 f(x) = _ x 1 -mx^ m-1 -1 = m . The function is defined as f(1) = 2m - 1 . For the function to be discontinuous at x=1 , the condition _ x 1 f(x) f(1) must hold. Therefore, m 2m - 1 . Solving for m : 1 2m - m m 1 . Answer: m 1