Limits, Continuity & Differentiability
Step Function Limits
Grade 12
Question:
<p>The value of $\lim_{x \to a} [2 - x + \sqrt{1 + x}]$, where $a \in [0, 1)$ and $[\cdot]$ denotes the greatest integer function is</p>
<p>(a) 3</p>
<p>(b) 2</p>
<p>(c) 1</p>
<p>(d) 4</p>
Step-by-Step Solution
Key Concept: Evaluate the expression inside the greatest integer function over the given interval. The greatest integer function is constant when its argument stays within the same integer interval.
<p>For $a \in [0, 1)$: when $x = a$, we have $2 - a + \sqrt{1 + a}$.</p><p>When $a = 0$: $2 - 0 + \sqrt{1} = 3$.</p><p>When $a \to 1^-$: $2 - 1 + \sqrt{2} = 1 + \sqrt{2} \approx 2.414$.</p><p>So $2 - a + \sqrt{1 + a} \in [3, 2.414)$ as $a$ increases from 0 to near 1.</p><p>Therefore, $[2 - a + \sqrt{1 + a}] = 2$ for $a \in (0, 1)$ and the limit is 2.</p>
Correct Answer: B