Limits, Continuity & Differentiability
Existence of Limits
Grade 12
Question:
<p>If \(f(x) = \frac{3x^2 + ax + a + 1}{x^2 + x - 2}\), which of the following can be correct?</p>
<p>(a) \(\lim_{x \to 1} f(x)\) exists: \(a = -2\)</p>
<p>(b) \(\lim_{x \to 0^+} f(x) = 1\)</p>
<p>(c) \(\cot^{-1}\left[\lim_{x \to 0^-} f(x)\right] = 1\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: For a limit to exist when the denominator has a zero, the numerator must also vanish at that point, requiring careful choice of parameter.
<p><strong>Solution:</strong> For $\lim_{x \to 1} f(x)$ to exist, the numerator must have factor (x-1) since denominator has (x-1) as a factor.</p><p>Set numerator at x=1: $3(1)^2 + a(1) + a + 1 = 0$</p><p>$3 + 2a + 1 = 0$</p><p>$a = -2$</p><p>∴ Option (a) is correct.</p>
Correct Answer: A