 Hello, when how are you all doing today? The question says, if fx is equal to mod of x plus 1 when the value of x is less than 0, when the value of x is equal to 0 then the value is 0 and mod of x minus 1 when the value of x is greater than 0. For what value or values of a does, of a does, limit x approaches to a, fx exists. Proceeding on with the solution, here we need to consider three cases. First of all, when x is less than 0, when x is equal to 0 and when x is greater than 0. When x is less than 0 then the function given to us is mod of x plus 1. When x is equal to 0 then the function is given to us as equal to 0 and when x is greater than 0 then the function which is given to us is mod of x minus 1. Now here, let us consider the limit x is equal to 0 from the left hand side. So x approaches to 0 from the left hand side we have the function as mod of x plus 1. Now we know that when the value of x is less than 0 then the value of mod x is minus x. So here we have the answer as minus x plus 1 or it can be written as 1 minus x. Now here the function is given to us as equal to 0 so we cannot proceed further. Here we have the limit as when x approaches to 0 from the right hand side we have mod of x minus 1. Now we know that when the value of x is greater than 0 then the value of mod of x is a positive x. So here the answer will be x minus 1. Now let this be the first, second and third equation. So clearly we can see that limit x approaches to 0 from the left hand side the function is not equal to the limit x approaches to 0 from the right hand side. Here we have 1 minus x and here we have x minus 1. So we can see that therefore limit x approaches to 0 the function does not exist at x is equal to 0. So that implies that the limit x approaches to a of the function will exist for all values of a when the value of a is not equal to 0 because here when x is not equal to 0 let us assume that x is equal to a. So the limit will exist for all a when it is not equal to 0 because at 0 limit of x approaches to 0 fx does not exist. So this is the required answer to the question that was given to us hope you understood have a nice day.