Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>limx→1+
(1−|x|+sin |1−x|) sin( π
2 [1−x])
|1−x|[1−x]
is equal to:
(1) −1
(2) 1
(3) D.N.E
(4) 0</p>
Step-by-Step Solution
Key Concept: General
<p><strong>1</strong>: As x \to 1+, |x| = x, |1 -x| = x -1, and [1 -x] = -1.</p><p><strong>2</strong>: Substitue: (1-x+sin(x-1)) sin(-\pi/2)</p> (x-1)(-1) = sin(x-1)-(x-1) x-1 .<p><strong>3</strong>: Let t = x -1. \lim_{t \to 0} sin t-t</p> t = \lim_{t \to 0} sin t t -1 = 1 -1 = 0.
Correct Answer: (4)