3a. Let f,g: ACR → R and c E A'. Assume that there is a 6> 0 such that f (x) = g(x) for all x € A such that 0 < x − c < 6. Show that if lim f (x) exist then limg (x) exists and x-c x-c lim f (x) x-c = limg(x). Argue that lim g(x) exists and x-c x-c In other words, lim f (x) depends only on the values of f (x) for x near c- this fact is often expressed by saying that limits are a "local property". x→a Notice that if lim f (x) = L exists then for € > 0, there is 0 < 6₁ ≤ 6, such that, for all x € A, if 0< x- c < 6₁, then f (x) - L < €. x-c Show that, for all x € A, if 0 < |x − c < 8₁, then [g (x) − L| < €. lim f (x) = limg(x). x-c x-C Explain the meaning of the expression that limits are a "local property".
3a. Let f,g: ACR → R and c E A'. Assume that there is a 6> 0 such that f (x) = g(x) for all x € A such that 0 < x − c < 6. Show that if lim f (x) exist then limg (x) exists and x-c x-c lim f (x) x-c = limg(x). Argue that lim g(x) exists and x-c x-c In other words, lim f (x) depends only on the values of f (x) for x near c- this fact is often expressed by saying that limits are a "local property". x→a Notice that if lim f (x) = L exists then for € > 0, there is 0 < 6₁ ≤ 6, such that, for all x € A, if 0< x- c < 6₁, then f (x) - L < €. x-c Show that, for all x € A, if 0 < |x − c < 8₁, then [g (x) − L| < €. lim f (x) = limg(x). x-c x-C Explain the meaning of the expression that limits are a "local property".
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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Please answer all parts. and include the picture of your solution. Thanks
Transcribed Image Text:3a.
Let f,g: A CR → R and c € A'. Assume that there is a d > 0 such that ƒ (x) = g(x) for
all x € A such that 0 < x − c < 8. Show that if lim f (x) exist then limg(x) exists and
X→C
Notice that if lim f (x)
X-C
0 < |x − c| < 8₁, then |ƒ (x) − L| < €.
lim f (x) = lim g(x).
x→C
X-C
In other words, lim f (x) depends only on the values of f (x) for x near c -
x-a
by saying that limits are a "local property".
=
x→C
L exists then for € > 0, there is 0 < 6₁ ≤ 6, such that, for all x € A, if
Argue that lim g (x) exists and
x→C
this fact is often expressed
Show that, for all x ¤ A, if 0 < |x − c| < 8₁, then |g (x) − L| < €.
lim f (x) = lim g(x).
X-C
X→C
Explain the meaning of the expression that limits are a "local property".
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