show that f(x) =|x-2| is continuous but not differentiable at x=2. Please reply!!URGENT!!!
a).|x-2| being a modulus function is continuous
b)For cheking differentiability, at x=2, we need to check if lim h=>0+ ( f(2+h) -f(2)/h) should not be equal to lim h=>0- ( f(2+h) -f(2)/h) or not
Now, lim h=>0+ f(2+h) -f(2)/h) = lim h=>0+ ( !2+h-2! -!2-2! ) /h = lim h=>0+ !h! /h = lim h=>0+ h/h =1 (as h= +ve hence !h!=h)
Similarly, lim h=>0- f(2+h) -f(2)/h) = lim h=>0- ( !2+h-2! -!2-2! ) /h = lim h=>0- !h! /h = lim h=>0- - h/h =-1 (as h= -ve hence !h!= -h)
so lim h=>0+ f(2+h) -f(2)/h) is equal to lim h=>0- f(2+h) -f(2)) /h