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Suppose there are three functions f.g, and h that satisfy the following inequalities h(x) < f(x) < g(x) for all such that 0 < x-c0 and lim h(x) = H, lim f(x) = F, and lim g(x) = G, for some real numbers H, F, and G. IIC I-C AIC We claim that H< F

Suppose there are three functions f.g, and h that satisfy the following inequalities h(x) < f(x) < g(x) for all such that 0 < x-c0 and lim h(x) = H, lim f(x) = F, and lim g(x) = G, for some real numbers H, F, and G. IIC I-C AIC We claim that H< F

Advanced Engineering Mathematics
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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Suppose there are three functions f, g, and h that satisfy the following inequalities h(x) < f(x) < g(x) for all z such that 0<x-c<p for some p > 0 and lim h(x) = H, lim f(x) = F, and lim g(x) = G, for some real numbers H, F, and G. 110 I-C x-C We claim that H< F<G Find a counterexample. That is, find an example that satisfies all the given conditions but does not satisfy the claim. Find a correct claim and prove it.
Transcribed Image Text:Suppose there are three functions f, g, and h that satisfy the following inequalities h(x) < f(x) < g(x) for all z such that 0<x-c<p for some p > 0 and lim h(x) = H, lim f(x) = F, and lim g(x) = G, for some real numbers H, F, and G. 110 I-C x-C We claim that H< F<G Find a counterexample. That is, find an example that satisfies all the given conditions but does not satisfy the claim. Find a correct claim and prove it.
1. Let's recall the Pinching (Squeeze) theorem (Theorem 2.5.1); Suppose there are three functions f, g, and h that satisfy the following inequalities on a given interval containing a constant c. h(x) ≤ f(x) ≤ g(x) for all x such that 0<x-c<p for some p>0 If the functions h and g have the same limits as x approaches c, then the function f also has the same limit. That is, if lim h(x) = lim g(x) = L, for some real number L, then lim f(x) = L. I-C Now, our goal is to generalize this theorem. The following is our initial incorrect attempt. You are going to fix it.
Transcribed Image Text:1. Let's recall the Pinching (Squeeze) theorem (Theorem 2.5.1); Suppose there are three functions f, g, and h that satisfy the following inequalities on a given interval containing a constant c. h(x) ≤ f(x) ≤ g(x) for all x such that 0<x-c<p for some p>0 If the functions h and g have the same limits as x approaches c, then the function f also has the same limit. That is, if lim h(x) = lim g(x) = L, for some real number L, then lim f(x) = L. I-C Now, our goal is to generalize this theorem. The following is our initial incorrect attempt. You are going to fix it.
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