Let f(x)= cases 5 e^ 1 / x +2 3-e^ 1 / x , & x 0 0, & x=0 cases Then at x=0, x f(x) and f(x) are respectively

Step-by-step solution

f(x)= array cc 5 e^ 1 / x +2 3-e^ 1 / x & x 0 0 & x=0 array . Then x f(x)= array cc x (5 e^ 1 / x +2 ) 3-e^ 1 / x & x 0 0 & x=0 array . Let us check continuity of x f(x) at x=0 aligned & L.H.L = _ h 0 (-h) [5 e^ -1 / h +2 ] 3-e^ -1 / h =0 & R.H.S = _ h 0 [55^ 1 h +2 ] 3-e^ 1 h =0 aligned x f(x)=0 x f(x) is continous on x=0 Let us check differentiability of f(x) at x=0 : aligned & L.H.D. = _ h 0 f(a-h)- f (0) -h & = _ h 0 5 e^ -1 / n +2 3-e^ -1 / n -0 -h aligned = _ h 0 - 1 h ( 5 e^ -1 / n +2 3-e^ -1 / h ) L.H.D is