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

The function f(x) = (√(x + 4) − 2)/x is defined for x ≠ 0, and f(0) = k. For what value of k is f continuous at x = 0?

k must be 1/4. Rationalising the numerator gives f(x) = 1/(√(x + 4) + 2) for x ≠ 0, whose limit as x tends to 0 is 1/(2 + 2) = 1/4. Continuity requires f(0) to equal this limit.

  1. A0
  2. B1/4Correct
  3. C1/2
  4. D4

Explanation

Multiply numerator and denominator by (√(x + 4) + 2). The numerator becomes (x + 4) − 4 = x, so f(x) = 1/(√(x + 4) + 2) for x ≠ 0. As x → 0 this tends to 1/(2 + 2) = 1/4. Continuity requires k = 1/4. The value 1/2 comes from forgetting to add the 2 in the denominator.

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