Paragraph: Let f(x)= array ll e^x-1 x , & x 0 0, & x=0 array . be a real valued function. Question: Which of the following statements is/are correct? 1. f(x) is right continuous at x=0 . 2. f(x) is discontinuous at x=1 . Select the correct answer using the code given below:

Step-by-step solution

For right hand continuity at x=0 aligned & aligned RHL & = _ x 0⁺ f(x)= _ h 0 f(0+h) & = _ h 0 e^h-1 h = _ h 0 (1+h+ h^2 2! + )-1 h & = _ h 0 h+ h^2 2! + h^3 3! + h & = _ h 0 1+ h 2! + h^2 3! + . .=1 aligned & array l f(0)=0 f(x) is not right continuous at x=0 . For discontinuity at x=1 array & RHL = _ x 1⁺ f(x)= _ h 0 f(1+h) aligned aligned & = _ h 0 e^ 1+h -1 1+h & = _ h 0 (1+(1+h)+ (1+h)^2 2 + . )-1 1+h & = _ h 0 (1+h)+ (1+h)^2 2! + (1+h)^3 3! + . (1+h) & = _ h 0 1+ (1+h) 2! + (1+h)^2 3! + . & =1+ 1 2! + 1 3! +