Definition: Let a E R and f : R {a} → R be a function. We say that f(x) → +∞ as x → aif and only if for every real number M there exists a positive d such that, for all real x withx - a| < 8 and x+ a, we have f (x) > M.Notice the similarities between this and Ross (1), page 160. The inequality |f(x) – L| <ɛ (thissays 'f(x) is close to L') has been replaced by f(x) > M (this says 'f(x) is large’.)2. Let g : R → R be a bounded function and define f : R {0} →R by1f(x) = g(x)+x2Using only the definition above, and without using any limit rules, prove that f (x) → +∞ as x → 0.

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Answered: 2. Let g : R → R be a bounded function…

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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2. Let g : R → R be a bounded function and define f : R\ {0} → R by 1 f(x) = g(x)+ x2 Using only the definition above, and without using any limit rules, prove that f(x) → +∞ as x → 0.