CA Foundation · Quantitative Aptitude · Sets, Relations and Functions, Limits and Continuity
Evaluate lim(x → 2) (x³ − 8)/(x² − 4).
The limit is 3. Factorising the numerator as (x − 2)(x² + 2x + 4) and the denominator as (x − 2)(x + 2), then cancelling x − 2, leaves (x² + 2x + 4)/(x + 2). Substituting x = 2 gives 12/4 = 3.
- A2
- B3Correct
- C4
- D12
Explanation
Factor: x³ − 8 = (x − 2)(x² + 2x + 4) and x² − 4 = (x − 2)(x + 2). Cancel (x − 2) to get (x² + 2x + 4)/(x + 2). At x = 2 this is 12/4 = 3. The option 12 is the numerator's derivative-type value 3x² taken without dividing by 2x = 4.
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