Limits, Continuity & Differentiability
Continuity at a Point
Grade 12
Question:
<p>If <span class="math">f(x) = \begin{cases} \frac{x^2+(a-2)x-2a}{x-2} & , x \neq 2 \\ 2 & , x = 2 \end{cases}</span> is continuous at <span class="math">x=2</span>, then <span class="math">a</span> is equal to</p>
<p>(a) <span class="math">0</span></p>
<p>(b) <span class="math">1</span></p>
<p>(c) <span class="math">-1</span></p>
<p>(d) <span class="math">2</span></p>
Step-by-Step Solution
Key Concept: Factor the numerator to cancel the problematic denominator, then use the continuity condition.
<p><strong>Solution:</strong> Factor the numerator: <span class="math">x^2+(a-2)x-2a = (x-2)(x+a)</span>. Then <span class="math">\lim_{x \to 2} \frac{(x-2)(x+a)}{x-2} = \lim_{x \to 2} (x+a) = 2+a</span>. For continuity at <span class="math">x=2</span>, we need <span class="math">2+a = 2</span>, so <span class="math">a = 0</span>.</p>
Correct Answer: a