Question

Let f(x) be a real valued function. Then its Left Hand Derivative (L.H.D.) : Lf′(a) = lim_{h→0} (f(a – h) – f(a))/(–h) Right Hand Derivative (R.H.D.) : Rf′(a) = lim_{h→0} (f(a + h) – f(a))/h Also, a function f(x) is said to be differentiable at x = a if its L.H.D. and R.H.D. at x = a exist and both are equal. For the function f(x) = {|x – 3|, x ≥ 1; x²/4 – 3x/2 + 13/4, x < 1 answer the following questions :

Find lim[x→0] f(x) and lim[x→1] f(x), where f(x) = { 2x+3, x ≤ 0; 3(x+1), x > 0 }

Find lim[x→1] f(x), where f(x) = { x²-1, x ≤ 1; -x²-1, x > 1 }

Evaluate lim[x→0] f(x), where f(x) = { |x|/x, x ≠ 0; 0, x = 0 }