(Degree Rule) Let D be an integral domain and f(x), g(x) E D[x]. Prove that deg (f(x) · g(x)) = deg f(x) + deg g(x). Show, by ex- ample, that for commutative ring R it is possible that deg f(x)g(x) < deg f(x) + deg g(x), where f(x) and g(x) are nonzero elements in R[x].

Q: x-3 5) Use Squeeze Theorem to prove that lim √x + x2 cos cos = 0. x-0 X 2-7--10

A: Step 1:To prove that limx→0​x4+x2​cos(x1​)=0 using the Squeeze Theorem, let's proceed with the…

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Q: Find the following limits, clearly showing all of your work.

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Q: use the following theorem to find £ ( 2 (1= (ost) Theorem: If £(f(t)) = F (s), then £ ( f(+)) = (*…

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Q: If f(x) and g(x) are arbitrary polynomials of degree at most 2, then the mapping (f,g) = f(-3)g(-3)…

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Q: If f(x) and g(x) are arbitrary polynomials of degree at most 1, then the mapping (f,g) = f(-2)g(-2)…

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Q: Let ƒ: R² → R² be an isomorphism where ([-])-[²] and f([:]) = [] = Find f ¹ƒ( [7¹]) X 0 2 2 0

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Q: 2) Evaluate Væ2 + 9 – 5 a) lim x -4 x+ 4 In(x +1) b) lim x 0 COS x – 1

A: Explanation of the answer is as follows

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Q: 10.2. Let f(x) = 3x³ + x* + x³ + 3x? + 2x +4 and g(x) = 2x³ + 3x² + x +1 be polynomials in Zs[x].…

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Q: Let F = {f ∈ C(R) : lim|x|→∞f(x) = 0 }. Show that F is closed in C(R

A: please  see the next step for solution

Q: - Give on homeomorphism./ example of continous function fiR-3R which is not o .//

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A: Given that 2x3+x2-13x+6 find the roots of the given polynomial . show that one of the root is 1/2.

Q: 22- 7x + 12 Use continuity to evaluate lim In Use that In r is continuous on (0, x - 4

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Q: 2) Evaluate Lim X + 2² (X, Y, Z) (0, 0, 0) + Z 1 or show that it exists NOT.

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Q: 2d

A: Definition of P2​(R): Space of polynomials of degree at most 2 with real coefficients.Introduction…

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A: Polynomial a is reducible.

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A: Dera Student,    As per our guidelines we can answer only first three subqustions. So kindly…

Q: In x Apply L'Hop rule. lim

A: We have to find the limit by using Lhospital rule.