3b. Let f: AR, where ACR and c E A' and lime-c f (x) = L. Then there is U CR, U- open, c E U, and there is M≥ 0, such that, for all x ≤ Un A, |ƒ (x)| ≤ M. That is, if f has limit at c then f is locally bounded at c. Use the assumption limx→c f (x) = L to show that there is 6 > 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,d), |f (x)| < 1 + |L|. ● Argue that there is UCR, U - open, c € U, and there is M≥ 0, such that, for all xEUNA, f (x)| ≤ M. What should we take for U and for M? Show that if c E A, then a € D (c,d), f (x)| ≤ max {1+ |L|, |ƒ (c)|} . 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?
3b. Let f: AR, where ACR and c E A' and lime-c f (x) = L. Then there is U CR, U- open, c E U, and there is M≥ 0, such that, for all x ≤ Un A, |ƒ (x)| ≤ M. That is, if f has limit at c then f is locally bounded at c. Use the assumption limx→c f (x) = L to show that there is 6 > 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,d), |f (x)| < 1 + |L|. ● Argue that there is UCR, U - open, c € U, and there is M≥ 0, such that, for all xEUNA, f (x)| ≤ M. What should we take for U and for M? Show that if c E A, then a € D (c,d), f (x)| ≤ max {1+ |L|, |ƒ (c)|} . 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?
Chapter3: Functions
Section3.3: Rates Of Change And Behavior Of Graphs
Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...
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Transcribed Image Text: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?
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