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CA Foundation · Quantitative Aptitude · Sets, Relations and Functions, Limits and Continuity

The function f is defined by f(x) = (x² − 5x + 6)/(x − 2) for x ≠ 2 and f(2) = k. For f to be continuous at x = 2, the value of k must be:

k must be −1. Factorising the numerator gives (x − 2)(x − 3), so f(x) simplifies to x − 3 for x ≠ 2. The limit at x = 2 is −1, and continuity requires the function value at 2 to equal this limit.

  1. A0
  2. B1
  3. C−1Correct
  4. D2

Explanation

For x ≠ 2, x² − 5x + 6 = (x − 2)(x − 3), so f(x) = x − 3. The limit as x → 2 is 2 − 3 = −1. Continuity requires f(2) = k = −1. Choosing 1 comes from a sign error.

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