89AE

STEP_BY_STEP SOLUTION Step-1 A continuous function can be formally defined as a f unction f : x y ,where the preimage of every open set in y is open in x. More concretely, a function f(x) in a single variable x is said to be continuous at point x if0 1. If f(x 0 is defined, so that x is 0n the domain of ‘ f’. 2. lim f(x) exists for x in the domain of f. x x0 3. lx x(x) = f( x ).0 0 Left continuous : lim f(x = f(a) , then f(x) is called a left continuous at x=a. xa Right continuous : lim f(x) = f(a) , then f(x) is called a right continuous at x=a. xa+ If , limf(x) = f(a) = lim +(x) , then f(x) is called a continuous function at x=a. xa xa If , f(x) is not continuous at x =a means , it is discontinuous at x=a. Step-2 x 4x +4x The given function is h(x) = , at x=1. x(x1) The given function is rational function , and it continuous for all values of x except x = / 0and 1. That is h( x) is discontinuous at x = 0 and 1. Note : If f(x) = p(x)is called a rational function , where q(x) = / 0. q(x) If q(x) =0 , then f(x) is discontinuous at these points. Step_3 a). Now , we need to check the discontinuity at x=0. The given function is a rational function , and as x approaches to zero, then the denominator value of h(x) is also approaches to zero. So, the numerator can be written as x 4x +4x = x(x -4x+4) 2 3 2 x 4x +4x x(x 4x+4) Hence , h(x) = x(x1) = x(x1) (x 4x+4) = (x1) , cancel out the like terms. (x2) = ……………….(1) (x1) 2 So, limh(x) = lim (x2) , from(1)...