Limits, Continuity & Differentiability
Limits with greatest integer function
Grade 12
Question:
<p>Let $f(x) = \begin{cases} |x - 2| + a^2 - 6a + 9, & x < 2 \\ 5 - 2x, & x \geq 2 \end{cases}$</p><p>If $\lim_{x \to 2} [f(x)]$ exists (where $[\cdot]$ represents the greatest integer function), the possible values $a$ can take is/are:</p>
<p>(a) $2$</p>
<p>(b) $\frac{5}{2}$</p>
<p>(c) $3$</p>
<p>(d) $\frac{7}{2}$</p>
Step-by-Step Solution
Key Concept: For the limit of the greatest integer function to exist at $x = 2$, the left and right limits of $f(x)$ must yield the same integer value.
<p>The correct answers are (b), (c), and (d).</p>
Correct Answer: B, C, D