If `f(y)={(((e^(2y)-1)*siny)/(y^(2)),", for "y ne 0),(4,", for "y=0):}`, then f(y) is

Options

A. discontinuous at y = 0
B. continuous at y = 0
C. not defined
D. None of the above

Answer

A

Explanation

Given, `f(y)={(((e^(2y)-1)siny)/(y^(2))"," , y ne 0),(4",", y =0):} `
Now `lim_(y to 0) f(y) = lim_( y to 0) ((e^(2y)-1))/(2y) xx 2 xx (siny)/(y)`
`=log e xx 2 xx 1 = 2 " but " f(0) = 4`
`therefore f(0) ne lim_( y to 0) f(y)`
Hence, f(y) is discontinuous at y = 0