3b.Let ƒ : A → R, where A CR and c € A' and limx→c f (x) = L. Then there is UCR, U -open, c E U, and there is M≥ 0, such that, for all x EUÑA, |ƒ (x)| ≤ M. That is, if ƒ has limitat c then f is locally bounded at c.Use the assumption lim-c f (x) L to show that there is d > 0, such that for all x € A, if0 < x − c < 6, then |ƒ (x)| < 1 + |L|.Show that, if c & A, then for all x ≤ D (c, 8),|ƒ (x)| < 1 + |L|.Argue that there is U CR, U - open, c € U, and there is M ≥ 0, such that, for allxe Un A, f (x)| ≤ M. What should we take for U and for M?Show that if c E A, then x E D (c, 8),|ƒ (x)| ≤ max {1+ |L|, |ƒ (c)|} .Argue that there is U CR, U open, c E U, and there is M≥ 0, such that, for allx ≤ UÑA, |ƒ (x)| ≤ M. What should we take for U and for M?

Given:The function , where , and and .Concept used:If has limit at , then is locally bounded at…

Answered: 3b. Let f: AR, where ACR and c E A' and…

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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Related questions

Question

3b.
Let ƒ : A → R, where A CR and c € A' and limx→c f (x) = L. Then there is UCR, U -
open, c E U, and there is M≥ 0, such that, for all x EUÑA, |ƒ (x)| ≤ M. That is, if ƒ has limit
at c then f is locally bounded at c.
Use the assumption lim-c f (x) L to show that there is d > 0, such that for all x € A, if
0 < x − c < 6, then |ƒ (x)| < 1 + |L|.
Show that, if c & A, then for all x ≤ D (c, 8),
|ƒ (x)| < 1 + |L|.
Argue that there is U CR, U - open, c € U, and there is M ≥ 0, such that, for all
xe Un A, f (x)| ≤ M. What should we take for U and for M?
Show that if c E A, then x E D (c, 8),
|ƒ (x)| ≤ max {1+ |L|, |ƒ (c)|} .
Argue that there is U CR, U open, c E U, and there is M≥ 0, such that, for all
x ≤ UÑA, |ƒ (x)| ≤ M. What should we take for U and for M?

Expert Solution

Step 1: Given and definition of limit

Given:

The function $f colon A rightwards arrow straight real numbers$ , where $A subset of straight real numbers$ , and $c element of A apostrophe$ and $limit as x rightwards arrow c of f open parentheses x close parentheses equals L$ .

Concept used:

If $f$ has limit at $c$ , then $f$ is locally bounded at $c$ .

Limit: If $limit as x rightwards arrow c of f open parentheses x close parentheses equals L$ then for every $epsilon greater than 0$ , $there exists space delta greater than 0$ such that for $open vertical bar f open parentheses x close parentheses minus L close vertical bar less than epsilon$ , whenever $0 less than open vertical bar x minus c close vertical bar less than delta.$