Find LhL and RHL of f(x)=|x|÷x …

Ex 5.1,8 teachoo.com Find all points of discontinuity off, where fis defined by 1x f(x) = if x = 0 10, if x = 0 Since we need to find continuity at of the function We check continuity for different values of x When x = 0 When x>0 When x<0 Since there are two different Case 1: When x = 0 f(x) is continuous at x = = 0 functions on the left & right of o, we take LHL & RHL. if L.H.L = R.H.L= f(0) if f lim f(x) lim f(x) = f(0) X-0- X-07 LHL at x → 0 RHL at x 0 lim f(x) = lim f(0 - h) X-0 h-0 lim f(x) = lim f(0 + h) h-0 x0+ = lim f(-h) h→0 = lim f(h) h→0 -h| = lim h→0 -h Th1 = lim h0 h h = lim h-h h = lim h-oh = lim -1 h-0 - lim 1 h→0 = 1 = -1 Since L.H.LR.H.L f(x) is not continuous at x = 0 Case 2: When x < 0 For x<0, f(x) = L x 1 (-x) f(x) = (As x < 0, x is negative) f(x) = -1 Since this constant It is continuous .. :: f(x) is continuous for x < 0 Case 3: When x > 0 For x>0, 1x f(x) - X х f(x) = X f(x) = 1 (As x > 0, x is positive) Since this constant It is continuous . :: f(x) is continuous for x > 0 Hence, only x = 0 is point is discontinuity. :: f is continuous for all real numbers except 0. Thus, f is continuous for x ER-{0}