Let `f(x)={{:(x[(1)/(x)]+x[x],if, x ne0),(0,if,x=0):}` (where [x] denotes the greatest integer function). Then the correct statement is/areA. Limit exists for `x=-1`.B. f(x) has a removable discontinuity at x = 1.C. f(x) has a non removable discontinuity at x = 2.D. f(x) is discontinuous at all positive integers.

Let `f(x)={{:(x[(1)/(x)]+x[x],if, x ne0),(0,if,x=0):}` (where [x] deno

1.	Let `f(x)={{:(x[(1)/(x)]+x[x],if, x ne0),(0,if,x=0):}` (where [x] denotes the greatest integer function). Then the correct statement is/areA. Limit exists for `x=-1`.B. f(x) has a removable discontinuity at x = 1.C. f(x) has a non removable discontinuity at x = 2.D. f(x) is discontinuous at all positive integers.
Answer» Correct Answer - A::B::C::D `f(1^(+))=underset(xrarr1^(+))(lim)(x[(1)/(x)]+x[x])` `=underset(xrarr1^(+))(lim)(x(0)+x(1))` `=1` `f(1^(-))=underset(xrarr1^(-))(lim)(x[(1)/(x)]+x[x])` `=underset(xrarr1^(-))(lim)(x(0)+x(1))` `=1` `f(2^(+))=underset(xrarr2^(+))(lim)(x[(1)/(x)]+x[x])` `=underset(xrarr2^(+))(lim)(x(0)+x(2))` `=4` `f(2^(-))=underset(xrarr2^(-))(lim)(x[(1)/(x)]+x[x])` `=underset(xrarr2^(-))(lim)(x(0)+x(2))` `=2` Obviously f(x) is discontinuous at all positive integers but at x = 1 it has removable discontinuity.