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

Let f(x) = (x² − 16)/(x − 4) for x ≠ 4 and f(4) = 6. Which statement about f at x = 4 is correct?

f has a removable discontinuity at x = 4. The limit of (x² − 16)/(x − 4) equals x + 4 = 8 as x approaches 4, but f(4) is given as 6. Since the limit exists and differs from the function value, continuity fails.

  1. Af is continuous because f(4) is defined
  2. BThe limit does not exist at x = 4
  3. Cf has a removable discontinuity at x = 4Correct
  4. Df has an infinite discontinuity at x = 4

Explanation

For x ≠ 4, f(x) = x + 4, so the limit as x tends to 4 is 8. But f(4) = 6, which is not equal to 8. The limit exists yet differs from the function value, so the discontinuity is removable by redefining f(4) = 8.

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