If f (x) = {mx² + n, x < 0; nx + m, 0 ≤ x ≤ 1; and nx³ + m, x > 1}. For what integers m and n does both limₓ→₀ f (x) and limₓ→₁ f (x) exists?

Solution:

The given function is, f (x) = {mx² + n, x < 0; nx + m, 0 ≤ x ≤ 1; and nx³ + m, x > 1}.

We will calculate the left hand and right hand limits at x = 0

ₓ→₀₋ f (x) = limₓ→₀ [mx² + n] = [m(0)² + n] = n

ₓ→₀₊ f (x) = limₓ→₀ [nx + m] = [n(0) + m] = m 

Thus, ₓ→₀ f (x) exists if m = n

Now, we will calculate the left hand and right hand limits at x = 1

lim_x→1^- f (x) = limₓ→₁ [nx + m] = [m(1)² + n] = m + n

lim_x→1⁺ f (x) = limₓ→₁ [nx³ + m] = [n(1)³ + m] = m + n

Therefore, lim_x→1^- f (x) = lim_x→1⁺ f (x) = limₓ→₁ f (x)

Thus, limₓ→₁ f (x) exists for any integral values of m and n

NCERT Solutions Class 11 Maths Chapter 13 Exercise 13.1 Question 32

If f (x) = {mx² + n, x < 0, nx + m, 0 ≤ x ≤ 1 and nx³ + m, x > 1} For what integers m and n does both limₓ→₀ f (x) and limₓ→₁ f (x) exists?

Summary:

limₓ→₀ f (x) exists if m = n and limₓ→₁ f (x) exists for any integral values of m and n