# 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 [mx^{2} + 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 [nx^{3} + 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) = {mx^{2} + n, x < 0, nx + m, 0 ≤ x ≤ 1 and nx^{3} + 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

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