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

Let f(x) = (x² − 4x + 3)/(x − 1) for x ≠ 1. The value that must be assigned to f(1) so that f is continuous at x = 1 is:

The value is −2. Factorising the numerator gives (x − 1)(x − 3), so f(x) equals x − 3 for x not equal to 1. The limit at 1 is −2, and f(1) must equal this limit for continuity.

  1. A2
  2. B−2Correct
  3. C0
  4. D4

Explanation

x² − 4x + 3 = (x − 1)(x − 3), so f(x) = x − 3 for x ≠ 1. The limit at x = 1 is 1 − 3 = −2. Defining f(1) = −2 makes f continuous. Choosing 2 comes from the sign error of using 3 − 1.

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