Limits, Continuity & Differentiability
Continuity with Floor Function
Grade 12
Question:
<p>If <span class="math">f(x) = \begin{cases} [x]-[-x] & , x \neq 2 \\ ; & , x = 2 \end{cases}</span> and <span class="math">f</span> is continuous at <span class="math">x=2</span>, where <span class="math">[\cdot]</span> denotes greatest integer function, then <span class="math">;</span> is</p>
<p>(a) <span class="math">-1</span></p>
<p>(b) <span class="math">0</span></p>
<p>(c) <span class="math">1</span></p>
<p>(d) <span class="math">2</span></p>
Step-by-Step Solution
Key Concept: Evaluate the greatest integer function on both sides of x=2 and apply continuity condition.
For $f(x)$ to be continuous at $x=2$, the limit of $f(x)$ as $x$ approaches 2 must exist and be equal to $f(2)$.
That is,
$$ \lim_{x \to 2} f(x) = f(2) $$
Given $f(2) = \;$, we must determine the value of $\lim_{x \to 2} f(x)$.
For $x \neq 2$, $f(x) = [x]-[-x]$.
By properties of the greatest integer function, the limit of this expression as $x$ approaches 2 is:
$$ \lim_{x \to 2} ([x]-[-x]) = 0 $$
Therefore, for $f(x)$ to be continuous at $x=2$, we must have:
$$ \; = 0 $$
Correct Answer: b