Limits, Continuity & Differentiability
Evaluation of Limits
Grade 12
Question:
<p>If <i>a</i> is the positive root of the equation <i>p</i>(<i>x</i>) = <i>x</i><sup>2</sup> − <i>x</i> − 2 = 0, then <code>lim</code><sub><i>x</i> → <i>a</i></sub> <span style="text-decoration: overline;">1 − cos(π<i>p</i>(<i>x</i>))</span>/<span style="text-decoration: overline;"><i>x</i><sup>2</sup> + <i>a</i> − 4</span> is equal to</p>
<p>(a) 1/2</p>
<p>(b) 1/2</p>
<p>(c) 3/2</p>
<p>(d) 3/2</p>
Step-by-Step Solution
Key Concept: Factor the quadratic to find the positive root, then substitute into the limit expression and apply L'Hôpital's rule or direct substitution.
<p><strong>Step 1:</strong> Given equation <i>p</i>(<i>x</i>) = <i>x</i><sup>2</sup> − <i>x</i> − 2 = (<i>x</i> − 2)(<i>x</i> + 1) having positive root <i>a</i>, so <i>a</i> = 2.</p><p><strong>Step 2:</strong> The positive root is <i>a</i> = 2.</p><p>∴ Answer is (c) 3/2.</p>
Correct Answer: C