Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>For each x ∈R, let [x] be the GIF. Then limx→0−x([x]+|x|) sin[x]
|x|
is equal to:
(1) −sin 1
(2) 0
(3) 1
(4) sin 1</p>
Step-by-Step Solution
Key Concept: General
<p><strong>1</strong>: As x \to 0-, |x| = -x and [x] = -1.</p><p><strong>2</strong>: Substitue: x(-1+(-x)) sin(-1)</p> -x = x(-1-x)(-sin 1) -x = (-1 -x) sin 1.<p><strong>3</strong>: Taking the limit as x \to 0-, we get -1 \cdot sin 1 = -sin 1.</p>
Correct Answer: (1)