(a) Let h: S→ R and c be a cluster point of S. Show that if lim h(x) = L = 0, then there exists some > 0 such that for all x € (S\ {c}) n (c-8, c+d), h(x) ‡ 0. 2-c (b) Let h: S→ R be continuous and c be a cluster point of S. Show that if h(c) # 0, then there exists some AC S such that c is a cluster point of A, h|₁ (x) ‡0 for all x ¤ A, and 1 lim (ALA(2)) - 1 lim (h(x)) h(c) I-C =
(a) Let h: S→ R and c be a cluster point of S. Show that if lim h(x) = L = 0, then there exists some > 0 such that for all x € (S\ {c}) n (c-8, c+d), h(x) ‡ 0. 2-c (b) Let h: S→ R be continuous and c be a cluster point of S. Show that if h(c) # 0, then there exists some AC S such that c is a cluster point of A, h|₁ (x) ‡0 for all x ¤ A, and 1 lim (ALA(2)) - 1 lim (h(x)) h(c) I-C =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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This problem contains a specal case of L'Hopitals rule
![### Educational Text on Limits and Continuity
#### Problem Statement:
(a) Let \( h: S \to \mathbb{R} \) and \( c \) be a cluster point of \( S \). Show that if \( \lim_{x \to c} h(x) = L \neq 0 \), then there exists some \( \delta > 0 \) such that for all \( x \in (S \setminus \{c\}) \cap (c - \delta, c + \delta) \), \( h(x) \neq 0 \).
(b) Let \( h: S \to \mathbb{R} \) be continuous and \( c \) be a cluster point of \( S \). Show that if \( h(c) \neq 0 \), then there exists some \( A \subset S \) such that \( c \) is a cluster point of \( A \), \( h|_{A}(x) \neq 0 \) for all \( x \in A \), and
\[
\lim_{x \to c} \left(\frac{1}{h|_{A}(x)}\right) = \frac{1}{\lim_{x \to c} (h|_{A}(x))} = \frac{1}{h(c)}
\]
#### Footnote:
1 Most trigonometric identities and inequalities have "geometric" proofs, so it doesn't count as "cheating" to use them to prove facts about calculus. See [Wikipedia link on Proofs of Trigonometric Identities](https://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities) for example.
#### Note:
This result allows us to "abuse notation". We get a slightly more general notion of Corollary 3.1.12.iv and write
\[
\lim_{x \to c} \left(\frac{1}{h(x)}\right) = \frac{1}{\lim_{x \to c} h(x)}
\]
even though strictly speaking, \( 1/h(x) \) might not be defined for all \( x \in S \).
This text discusses limits in calculus, providing conditions under which a function does not equal zero within a specified interval, along with how such properties extend to continuous functions. The focus is on understanding how these mathematical limits behave and are manipulated formally.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97e612ef-1556-436b-b62c-352b280e9e69%2Fdd026aa6-ae83-4ccd-afa7-454287c5c937%2F0qrg5ep_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Text on Limits and Continuity
#### Problem Statement:
(a) Let \( h: S \to \mathbb{R} \) and \( c \) be a cluster point of \( S \). Show that if \( \lim_{x \to c} h(x) = L \neq 0 \), then there exists some \( \delta > 0 \) such that for all \( x \in (S \setminus \{c\}) \cap (c - \delta, c + \delta) \), \( h(x) \neq 0 \).
(b) Let \( h: S \to \mathbb{R} \) be continuous and \( c \) be a cluster point of \( S \). Show that if \( h(c) \neq 0 \), then there exists some \( A \subset S \) such that \( c \) is a cluster point of \( A \), \( h|_{A}(x) \neq 0 \) for all \( x \in A \), and
\[
\lim_{x \to c} \left(\frac{1}{h|_{A}(x)}\right) = \frac{1}{\lim_{x \to c} (h|_{A}(x))} = \frac{1}{h(c)}
\]
#### Footnote:
1 Most trigonometric identities and inequalities have "geometric" proofs, so it doesn't count as "cheating" to use them to prove facts about calculus. See [Wikipedia link on Proofs of Trigonometric Identities](https://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities) for example.
#### Note:
This result allows us to "abuse notation". We get a slightly more general notion of Corollary 3.1.12.iv and write
\[
\lim_{x \to c} \left(\frac{1}{h(x)}\right) = \frac{1}{\lim_{x \to c} h(x)}
\]
even though strictly speaking, \( 1/h(x) \) might not be defined for all \( x \in S \).
This text discusses limits in calculus, providing conditions under which a function does not equal zero within a specified interval, along with how such properties extend to continuous functions. The focus is on understanding how these mathematical limits behave and are manipulated formally.
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