Working directly from definition, prove that if zn and w, are se- quences of complex numbers with lim zn = 4+ 3i, and lim wn = 4 – 3i, then lim z, · w, = 25. (You may use the fact that convergent complex se- n00 quences are bounded.)

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A: on solving this we get the

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A: Given: limn→∞ 6n2-2n412n3+2n2-17n   Property (i) limn→∞f(x)g(x)=limn→∞ f(x)limn→∞ g(x)    ,…

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A:

Q: a) Approximate the area under graph (a) of f(x)=1/x² over the interval [1,9] by computing the area…

A: To estimate the area under the curve in the given interval using the given rectangles.

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A: Given that Pz=a0+a1z+a2z2+⋯+anzn be a polynomial of degree n.Then, an≠0. We have to show that 1P(z)…

Q: Solve if w be cube root of unity then (1+w)(1+w^2)(1+w^4)(1+w^8).....2nfactor=1

A:

Q: can you explain how you got (x-2)=(t-3) and can you explain how you got the new limits because the…

A: Given:

Q: lim b300 مام 3 x (x+2) = 3 (x+2) 3 2 X + -3/1/13 (x+2) A X 3 2 x B (x+2) 1₁. 3 2(x+2) 3 x2 + 2x da…

A: The objective is to find the integration.

Q: 6n2 + 3n = 3 2n2 – 5 Prove: lim

A:

Q: How many zeros does f(x) = a(x - 2)^2 + 5 have if a > 0? a)1 b)2 c) impossible to determine d)0

A:

Q: Compute the following limit (if it exists). iz + 1 lim :→i 22 + 1 Hìnt: Write the denominator z2…

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Q: lim (z,y)→(0,0) (3 + (x² + y²) In(x² + y?)) 00 does not exist O 30 1,

A:

Q: 9. Find bijective functions from R to R a) f (x) = 2x + 1 b) f(x) = x² + 1 c) f (x) = x³ %3D x2+1 d)…

A: As per our guideline we are allowed to do 1 question or maximum 3 subparts at a time. Please post…

Q: 3x + 2x + x+ 1 Let Ax) = What types of asymptotes does Ax) have? Explain. .2 x* +x-2

A: To find the asymptotes of f(x).

Q: Determine the domain of a) {x|x=-1, xER} b) {x|x+1, xR} c) {x|x*-LXR} d) {x\xR} 1 f(x) = x+1

A:

Q: 2) Prove that: d? L„0) = n;(n – 1) %3D dx²

A: Consider the given information: Let Laguerre polynomial is given by, Lnx=∑k=0n(-1)kk!nkxk…

Q: Vx? + 100 – 10 lim x² .2

A:

Q: 7. Let a be a nonzero real number. Evaluate lim o f (x) and lim,→ f(x), where -00 f (x) = a.x° – 3x*…

A: Given problem is :

Q: Find the following limits. You may use any theorem we have proved, such as Arithmetic of Limits.…

A:

Q: x? — х -2 lim 2 x +3x + 2 X→-1 X

A: First we factories then take limit.

Q: Find a monic polynomial of least degree in R[x] that has 1-i and 2i as roots.

A:

Q: Theorem 6 If (b+ f) > (c+r) and (d+g) > (e + s), then the necessary and sufficient condition for…

A: Given: →b+f>c+r⇒b+f-c+r>0....1→d+g>e+s⇒d+g-e+s>0....2 To prove:…

Q: (b) Assume C and D are chosen at random from {1,2., n}. Let pn be the probability that C + D is a…

A: Given: C and D are chosen at random from {1, . . . , n}. And pnis the probability that C + D is a…

Q: Vx+1-2 b) lim,

A:

Q: Show that the polynomial z5 + 14z + 2 has four zeros in the annulus </2 < 2.

A: Given

Q: 3. F(x) = n(1+t)dt Prove that the function f(x) = cor In(1+z) is Riemann integrable on (0,27]

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A:

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A: To Find: Splitting field of x2-22x+3=0 over ℚ2.

Q: Compute limnon (√4n² +3-2n) and justify your calculations.

A: Introduction: In mathematics, the limit of a function is a fundamental concept in calculus and…

Q: Can you please help with me (a)

A:

Q: het f be a bounded real-valued funtim on TR" Define (Mst)(a) = Aup {tla): Ix-al <8$ %3D (mg t)(a) -…

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Q: Prove that Using Cauchy's Residue Theorem 0 sin² (0) a + bcos (0) -d0 = 2π 62 -[a- √a² - 6²]

A:

Q: Explain theorem of Wronskians of Solutions?

A: To Find : Define the Theorem of Wronskians of solution