Question 30 of 70advanced🔧 ApplyShort Answer3 marks

If f(x)={mx2+n,x<0nx+m,0≤x≤1nx3+m,x>1f(x) = \begin{cases} mx^2 + n, & x < 0 \\ nx + m, & 0 \leq x \leq 1 \\ nx^3 + m, & x > 1 \end{cases}. For what integers mm and nn does both lim⁡x→0f(x)\lim_{x \to 0} f(x) and lim⁡x→1f(x)\lim_{x \to 1} f(x) exist?

Correct Answer

m = n, where m and n are any integers.

Exercise: EXERCISE 12.1 | Q: 32 | (Page Page 22)
For More Understanding

Explanation

This question tests the concept that for a limit to exist at a point, the left-hand limit (LHL) must equal the right-hand limit (RHL). The student needs to apply this condition at both x = 0 and x = 1 for the piecewise function.

Solution Steps

  1. Step 1: For limit to exist at x = 0, equate LHL and RHL. LHL: lim(x→0⁻)(mx² + n) = n. RHL: lim(x→0⁺)(nx + m) = m. So, n = m.

  2. Step 2: For limit to exist at x = 1, equate LHL and RHL. LHL: lim(x→1⁻)(nx + m) = n + m. RHL: lim(x→1⁺)(nx³ + m) = n + m. This condition is always satisfied.

  3. Step 3: Combining both conditions, the only requirement is m = n. Since m and n must be integers, they can be any equal integers (e.g., m = n = 0, 1, -1, 2, etc.).

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