Limits, Continuity & Differentiability
Limit Properties
Grade 12
Question:
<p>If \(\lim_{x \to a} f(x) = \lim_{x \to a} [f(x)]\) (where [·] denotes the greatest integer function) and <i>f</i>(<i>x</i>) is non-constant continuous function, then</p>
<p>(a) \(\lim_{x \to a} f(x)\) is an integer</p>
<p>(b) \(\lim_{x \to a} f(x)\) is non-integer</p>
<p>(c) <i>f</i>(<i>x</i>) has local maximum at <i>x</i> = <i>a</i></p>
<p>(d) <i>f</i>(<i>x</i>) has local minimum at <i>x</i> = <i>a</i></p>
Step-by-Step Solution
Key Concept: The limit equals its floor only when the limit is an integer value
<p>For a non-constant continuous function, if $\lim_{x \to a} f(x) = \lim_{x \to a} [f(x)]$, then the limit must be an integer, since the greatest integer function equals the function only at integer values.</p>
Correct Answer: A