<p>Let f(x) =
(
sin x
x ∈Z
0
x /∈Z and g(x) =
x2 + 1
x ̸= 0, 2
4
x = 0
5
x = 2
, then:</p>
Step-by-Step Solution
Key Concept: General
<p><strong>1</strong>: Check (A): For x \to 0, x is a small non-integer. Thus f(x) = 0. As x approaches 0,</p> f(x) is identically 0. So g(f(x)) = g(0) = 4. Correct.<p><strong>2</strong>: Check (B): As x \to 0, g(x) = x2 + 1 \to 1. However, for x ̸= 0, g(x) > 1 and is not an</p> integer. Thus f(g(x)) = 0. Correct.<p><strong>3</strong>: Check (C): As x \to 1, g(x) = x2 + 1 \to 2. For x near 1, g(x) is near 2 but not exactly</p> 2 (except at x = 1). In the deleted neighborhood, g(x) is non-integer, so f(g(x)) = 0. Correct.
Correct Answer: (A, B, C)