From the Table 5 2 we deduce that the value of f x may be

Last updated: 10/10/2023

From the Table 5 2 we deduce that the value of f x may be made smaller than any given number by choosing a negative real number very close to 0 In symbols we write lim f x 0 x 0 to be read as the left hand limit of f x at 0 is minus infinity Again we wish to emphasise that is NOT a real number and hence the left hand limit of fat 0 does not exist as a real number The graph of the reciprocal function given in Fig 5 3 is a geometric representation of the above mentioned facts 110 MATHEMATICS Rationalised 2023 24 f x lim f x lim x 2 c 2 f c X C X C Thus fis continuous at all points x 1 Case 3 If c 1 then the left hand limit of fat x 1 is Example 10 Discuss the continuity of the function f defined by x 2 if x 1 x 2 if x 1 lim f x lim x 2 1 2 3 x l x l The right hand limit of fat x 1 is f x 2 2 lim f x lim x 2 1 2 1 x 1 x 1 1 1 2 2 lim f x lim x 2 1 2 3 x 1 The right hand limit of fat x 1 is lim f x lim x 2 1 2 1 x 1 x 1 Since the left and right hand limits of fat x 1 Solution The function fis defined at all points of the real line Case 1 If c 1 then f c c 2 Therefore lim f x lim x 2 c 2 x c Thus fis continuous at all real numbers less than 1 Case 2 If c 1 then f c c 2 Therefore 3 1 3 3 2 1 2 2 1 1 1 3 3 Y X C 1 O X 1 2 3 Y Fig 5 3 ERT publiced 2 1 2 Example 11 Find all the points of discontinuity of the function f defined by x 2 if x 1 0 if x 1 x 2 if x 1 Solution As in the previous example we find that f is continuous at all real numbers x 1 The left hand limit of fat x 1 is 1 3 Since the left and right hand limits of fat x 1 do not coincide f is not continuous at x 1 Hence Fig 5 4 x 1 is the only point of discontinuity off The graph of the function is given in Fig 5 4 2 1 1 1 O 3 1 3 X 1 O 1 1 X F X 2 3