Limits, Continuity & Differentiability
Limits of Step Functions
Grade 12
Question:
<p>If <span class="math">f(x) = \begin{cases} A \sin[x] & \text{for } [x] \geq 0 \\ 0 & \text{for } [x] < 0 \end{cases}</span>, where $[x]$ denotes the greatest integer less than or equal to $x$, then $\lim_{x \to 0} f(x)$ equals</p>
<p>(a) 1</p>
<p>(b) 0</p>
<p>(c) -1</p>
<p>(d) doesn't exist</p>
Step-by-Step Solution
Key Concept: Evaluate one-sided limits using the greatest integer function definition. When $x$ approaches 0 from the left, $[x] = -1$ making $f(x) = 0$ by the second case.
<p>As $x \to 0^+$, $[x] = 0$, so $f(x) = A \sin(0) = 0$.</p><p>As $x \to 0^-$, $[x] = -1$, so $f(x) = 0$.</p><p>Both one-sided limits equal 0, so $\lim_{x \to 0} f(x) = 0$.</p>
Correct Answer: D