(a) Let f: [0, ∞) → R be defined by f(x) = √x. Show that lim f(x) ce [0, ∞). Is f a continuous function? 24x = √e for all (Remark: You may use the fact that 0 ≤ a < b if and only if √a < √b. As a hint on how to play the & games, look at the proof of Proposition 2.2.6 in the textbook.) (b) Let f: R → R be defined by f(x) := cos(x). Show that lim f(x) = cos(c) for all c € R. Is f a continuous function? X-C (Remark: You may use trigonometric identities here, and the fact that |sin(x)| ≤ |x|, and |sin(x)| ≤ 1 for all x € R. See Example 3.2.6 in the textbook for the necessary algebra; however, you will need explain all of the steps of the proof to receive credit.)
(a) Let f: [0, ∞) → R be defined by f(x) = √x. Show that lim f(x) ce [0, ∞). Is f a continuous function? 24x = √e for all (Remark: You may use the fact that 0 ≤ a < b if and only if √a < √b. As a hint on how to play the & games, look at the proof of Proposition 2.2.6 in the textbook.) (b) Let f: R → R be defined by f(x) := cos(x). Show that lim f(x) = cos(c) for all c € R. Is f a continuous function? X-C (Remark: You may use trigonometric identities here, and the fact that |sin(x)| ≤ |x|, and |sin(x)| ≤ 1 for all x € R. See Example 3.2.6 in the textbook for the necessary algebra; however, you will need explain all of the steps of the proof to receive credit.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Prove the following, using the ε-δ definition of the limit of a function:
Transcribed Image Text:### Mathematical Analysis Problems
#### (a) Function Continuity and Limits
**Problem:**
Let \( f : [0, \infty) \to \mathbb{R} \) be defined by \( f(x) := \sqrt{x} \). Show that \(\lim_{x \to c} f(x) = \sqrt{c}\) for all \( c \in [0, \infty) \). Is \( f \) a continuous function?
**Remark:**
You may use the fact that \( 0 \leq a < b \) if and only if \( \sqrt{a} < \sqrt{b} \). As a hint on how to play the \( \varepsilon \) games, look at the proof of Proposition 2.2.6 in the textbook.
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#### (b) Trigonometric Function Continuity and Limits
**Problem:**
Let \( f : \mathbb{R} \to \mathbb{R} \) be defined by \( f(x) := \cos(x) \). Show that \(\lim_{x \to c} f(x) = \cos(c)\) for all \( c \in \mathbb{R} \). Is \( f \) a continuous function?
**Remark:**
You may use trigonometric identities here, and the fact that \( |\sin(x)| \leq |x| \), and \(|\sin(x)| \leq 1\) for all \( x \in \mathbb{R} \). See Example 3.2.6 in the textbook for the necessary algebra; however, you will need to explain all of the steps of the proof to receive credit.
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