1. In this exercise, we will explore how the limit laws will not apply if the point at which the two functions being considered is such that one or both of the limits of those functions do not exist. A. Find a pair of example functions f: R → R and g: R → R for which lim (f(x) + g(x)) = 0 but neither lim f(x) nor lim g(x) exists. (Hint: jumps that balance each other out?) x→2 x→2 x-2 B. Find a pair of example functions f: R → R and g: (-∞, 2) U (2, ∞)→ R for which lim f(x) = 0 the value of lim (f(x) · g(x)) equals 1. (Hint: x-2 x-2 notice the domain of function g.) C. By choosing a different function g in part B, for any real number c, we could find a pair of functions for which lim f(x) = 0 and the value of x-2 lim (f(x) · g(x)) equals c. Explain. x-2
1. In this exercise, we will explore how the limit laws will not apply if the point at which the two functions being considered is such that one or both of the limits of those functions do not exist. A. Find a pair of example functions f: R → R and g: R → R for which lim (f(x) + g(x)) = 0 but neither lim f(x) nor lim g(x) exists. (Hint: jumps that balance each other out?) x→2 x→2 x-2 B. Find a pair of example functions f: R → R and g: (-∞, 2) U (2, ∞)→ R for which lim f(x) = 0 the value of lim (f(x) · g(x)) equals 1. (Hint: x-2 x-2 notice the domain of function g.) C. By choosing a different function g in part B, for any real number c, we could find a pair of functions for which lim f(x) = 0 and the value of x-2 lim (f(x) · g(x)) equals c. Explain. x-2
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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![1. In this exercise, we will explore how the limit laws will not apply if the point at which the two functions being considered is such that one or both of the limits of those functions do not exist.
A. Find a pair of example functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to \mathbb{R} \) for which \( \lim_{x \to 2} (f(x) + g(x)) = 0 \) but neither \( \lim_{x \to 2} f(x) \) nor \( \lim_{x \to 2} g(x) \) exists. *(Hint: jumps that balance each other out?)*
B. Find a pair of example functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: (-\infty, 2) \cup (2, \infty) \to \mathbb{R} \) for which \( \lim_{x \to 2} f(x) = 0 \) the value of \( \lim_{x \to 2} (f(x) \cdot g(x)) \) equals 1. *(Hint: notice the domain of function g.)*
C. By choosing a different function g in part B, for any real number \( c \), we could find a pair of functions for which \( \lim_{x \to 2} f(x) = 0 \) and the value of
\[ \lim_{x \to 2} (f(x) \cdot g(x)) \]
equals \( c \). Explain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe8612948-bcdc-496b-a0d4-ef3f637bada0%2F3aebb69f-154e-4e06-ae71-6a3fe598e946%2F0krrzyp_processed.png&w=3840&q=75)
Transcribed Image Text:1. In this exercise, we will explore how the limit laws will not apply if the point at which the two functions being considered is such that one or both of the limits of those functions do not exist.
A. Find a pair of example functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R} \to \mathbb{R} \) for which \( \lim_{x \to 2} (f(x) + g(x)) = 0 \) but neither \( \lim_{x \to 2} f(x) \) nor \( \lim_{x \to 2} g(x) \) exists. *(Hint: jumps that balance each other out?)*
B. Find a pair of example functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: (-\infty, 2) \cup (2, \infty) \to \mathbb{R} \) for which \( \lim_{x \to 2} f(x) = 0 \) the value of \( \lim_{x \to 2} (f(x) \cdot g(x)) \) equals 1. *(Hint: notice the domain of function g.)*
C. By choosing a different function g in part B, for any real number \( c \), we could find a pair of functions for which \( \lim_{x \to 2} f(x) = 0 \) and the value of
\[ \lim_{x \to 2} (f(x) \cdot g(x)) \]
equals \( c \). Explain.
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