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

Let f(x) = (x² − 4)/(x − 2) for x ≠ 2, and f(2) = 5. Which statement about f at x = 2 is correct?

f has a removable discontinuity at x = 2. The limit of (x² − 4)/(x − 2) is 4, but f(2) is defined as 5. Since the limit and the function value differ, the function is not continuous there.

  1. Af is continuous because f(2) exists
  2. Bf has a removable discontinuity, since the limit is 4 but f(2) = 5Correct
  3. Cf has an infinite discontinuity at x = 2
  4. Df is continuous because the limit equals 5

Explanation

For x ≠ 2, f(x) = x + 2, so the limit as x → 2 is 4. But f(2) = 5, which differs from the limit. The limit exists but does not equal the function value, so the discontinuity is removable. Redefining f(2) = 4 would make it continuous.

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