Limits, Continuity & Differentiability
Limits with Greatest Integer Function
Grade 12
Question:
<p>The value of $\lim_{x \to 1} \{1 + x + [x-1] + [1-x]\}$ (where $[\cdot]$ denotes the greatest integer function) is</p>
<p>(a) $-1$</p>
<p>(b) Doesn't exist</p>
<p>(c) $1$</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: The greatest integer function is discontinuous at integer points; check left and right hand limits separately.
<p>As $x \to 1$, we need to find both left and right hand limits using the greatest integer function properties.</p><p>For $x$ slightly less than 1: $[x-1] = -1$ and $[1-x] = 0$</p><p>For $x$ slightly greater than 1: $[x-1] = 0$ and $[1-x] = -1$</p><p>Since LHL $\neq$ RHL, the limit doesn't exist.</p>
Correct Answer: B