Limits, Continuity & Differentiability
General
Grade 12
Question:
<p>Let f(x) = 1−x(1+|1−x|)
|1−x|
cos
1
1−x
for x ̸= 1. Find LHL and RHL at x = 1.</p>
Step-by-Step Solution
Key Concept: General
<p>For x \to 1-, put h = 1 -x > 0. Then</p> f(x) = 1 -(1 -h)(1 + h) h cos 1 h = h cos 1 h \to 0. So the left-hand limit exists and equals 0. For x \to 1+, put h = x -1 > 0. Then f(x) = 1 -(1 + h)(1 + h) h cos -1 h = (-2 -h) cos 1 h , which oscillates and has no limit.
Correct Answer: (A, C)