Consider the function f(x)= array ccc x -2 x & If & x 2 3 & If & x= 2 array . Which is continuous at x= 2 , where is a constant. What is the value of ?

Step-by-step solution

f(x)= array ccc x -2 x & if & x 2 3 & if & x= 2 array . For continuity at x = 2 L.H.L. = _ x ⁻ 2 f(x)= _ x ⁻ 2 x -2 x Put x = 2 - h where x 2 , then h 0 aligned & L.H.L. = _ h 0 ( 2 -h ) -2 ( 2 -h ) = _ h 0 h - +2 h & [ ( 2 - )= ] & = _ h 0 2 h 2 h = 2 _ h 0 h h & = 2 1= 2 [ _ 0 =1 ] & and R.H.L. = _ x ⁺ 2 f(x) _ x ⁺ 2 x -2 aligned Put x = 2 + h when x 2 , then h 0 R.H.L. = _ h 0 ( 2 +h ) -2 ( 2 +h ) = _ h 0 (- h) - -2 h array ll = _ h 0 - h -2 h & [ ( 2 + )=- ] = 2 _ h 0 h h = 2 & ( _ 0 =1 ) array Also, f ( 2 )=3