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Let g: A → R be a function, and suppose that f: A → R is a bounded function. (So, there exists MER with M >0 so that for all r A we have f(x)| ≤ M.) Let c be a limit point of A. Prove that if lim a(r) = 0 then lim a(r) f(x) = 0.

Let g: A → R be a function, and suppose that f: A → R is a bounded function. (So, there exists MER with M >0 so that for all r A we have f(x)| ≤ M.) Let c be a limit point of A. Prove that if lim a(r) = 0 then lim a(r) f(x) = 0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Let \( g: A \to \mathbb{R} \) be a function, and suppose that \( f: A \to \mathbb{R} \) is a bounded function. (So, there exists \( M \in \mathbb{R} \) with \( M > 0 \) so that for all \( x \in A \) we have \( |f(x)| \leq M \).) Let \( c \) be a limit point of \( A \). Prove that if \(\lim_{x \to c} g(x) = 0\) then \(\lim_{x \to c} g(x)f(x) = 0\).
Transcribed Image Text:Let \( g: A \to \mathbb{R} \) be a function, and suppose that \( f: A \to \mathbb{R} \) is a bounded function. (So, there exists \( M \in \mathbb{R} \) with \( M > 0 \) so that for all \( x \in A \) we have \( |f(x)| \leq M \).) Let \( c \) be a limit point of \( A \). Prove that if \(\lim_{x \to c} g(x) = 0\) then \(\lim_{x \to c} g(x)f(x) = 0\).
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