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x2+yx , JX= e-1/(2+y2)=0x-,f(e) f(j)fxy-xy9. Suppose f: R" R" is continuous and Xo is arbitrary. Define a sequence by Xk = 1, 2, .. .. Prove that if x a, then f (a) a. We say a is a fixed point of f.= f(xk-1),10. Use Exercise 9 to find the limit of each of the following sequences of points in R, presuming itexists.1(c) xo 1, xk = 1 +Xk-1V2Xk-1*(a) xo= 1, xk21*(d) xo 1, xk = 1 T 1 +Xk-1Xk-1+2(b) xo 5, x =1Xk-111. Give an example of a discontinuous function f: R - R having the property that for every c e Rthe level set f({c}) is closed.

Question

Problem 9 is included in this picture.

I only need help for problem 10c. Thank you very much!

x2+yx , J
X
= e-1/(2+y2)
=0
x-,f
(e) f
(j)f
xy
-x
y
9. Suppose f: R" R" is continuous and Xo is arbitrary. Define a sequence by X
k = 1, 2, .. .. Prove that if x a, then f (a) a. We say a is a fixed point of f.
= f(xk-1),
10. Use Exercise 9 to find the limit of each of the following sequences of points in R, presuming it
exists.
1
(c) xo 1, xk = 1 +
Xk-1
V2Xk-1
*(a) xo= 1, xk
2
1
*(d) xo 1, xk = 1 T 1 +Xk-1
Xk-1
+
2
(b) xo 5, x =
1
Xk-1
11. Give an example of a discontinuous function f: R - R having the property that for every c e R
the level set f({c}) is closed.
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x2+yx , J X = e-1/(2+y2) =0 x-,f (e) f (j)f xy -x y 9. Suppose f: R" R" is continuous and Xo is arbitrary. Define a sequence by X k = 1, 2, .. .. Prove that if x a, then f (a) a. We say a is a fixed point of f. = f(xk-1), 10. Use Exercise 9 to find the limit of each of the following sequences of points in R, presuming it exists. 1 (c) xo 1, xk = 1 + Xk-1 V2Xk-1 *(a) xo= 1, xk 2 1 *(d) xo 1, xk = 1 T 1 +Xk-1 Xk-1 + 2 (b) xo 5, x = 1 Xk-1 11. Give an example of a discontinuous function f: R - R having the property that for every c e R the level set f({c}) is closed.

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check_circleAnswer
Step 1

Given:

-1+
-I
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-1+ -I

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Step 2

To find: The limit of the sequence.

Answer:

Let the sequence be defined as x = f(x,-)
Assume the limit of the sequence to be a
Then, by theorem in exercise 9, we have f(a)= a. Thus, it gives
1
1+- a
a+1a2
a2-a-1 0
help_outline

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Let the sequence be defined as x = f(x,-) Assume the limit of the sequence to be a Then, by theorem in exercise 9, we have f(a)= a. Thus, it gives 1 1+- a a+1a2 a2-a-1 0

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Step 3

Solve for a....

a2-a-1 0
-(-1)-1)()-1)
2(1)
lt5
2
For x1, the sequence is approaching towards a positive number
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a2-a-1 0 -(-1)-1)()-1) 2(1) lt5 2 For x1, the sequence is approaching towards a positive number

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