The right hand and left hand limit of the function are respectively. f(x)= array cc e^ 1 / x -1 e^ 1 / x +1 , & if x 0 0, & if x=0 array .

Step-by-step solution

We have, f(x)= array cc e^ 1 / x -1 e^ 1 / x +1 , & if x 0 0, & x=0 array . aligned Right hand limit &= _ x 0⁺ f(x)= _ h 0 f(0+h) &= _ h 0 f(h)= _ h 0 e^ 1 / h -1 e^ 1 / h +1 &= _ h 0 1-e^ -1 / h 1+e^ -1 / h & 1-0 1+0 1 Left hand limit =& _ x 0⁻ f(x)= _ h 0 f(0-h) =& _ h 0 f(-h) aligned = _ h 0 e^ -1 / h -1 e^ -1 / h +1 = _ h 0 ( e^ 1 1 / h -1 1 e^ 1 / h +1 )= 0-1 0+1 =-1