Use Equation (11.3.14) to prove that the Hermite polynomials satisfy |e* HN (x) HM (x)dx = 0, M # N. [Hint: Follow the steps taken in proving orthogo- nality of the Legendre polynomials. You will need to recall that lim e-x?, P(x) = 0, for any polynomial p.]

Q: B. Prove the fillang tim Jn2 +n 4- (6) lim n-700 Lvntl trn

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Q: Show that (2o + h)" – 2ö = nz- | n-1 lim h→0 h for any zo E C.

A: To show: limh→0z0+hn-z0nh=nz0n-1for any z0∈ℂ

Q: Given the characteristic polynomial (r – n) - (r- r2) = 0, find the associated re- currence…

A: The required recurrence relation is, an + (r1r2)an-2 =  (r1+r2)an-1    The detailed solution is as…

Q: 5. If b > 0, prove that lim = 0. %3D 1+nb

A: as per our guidelines we are supposed to answer?️ only one question. Kindly repost other question as…

Q: 43()3De"fur all regl numbras x.find Lim g D4-ズ

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Q: :f and the functions fn are all monotone increas- If limn0. fn ing, must f be monotone increasing?…

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Q: evaluate the limit. lim e2 'i+ In(t + 1)j + 4k

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Q: Pick 8 of 10: Prove or provide a counterexample. lim Sn =0 if and only if lim s,=0

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Q: In formally proving that lim (z² + x) = 42, let e > 0 be arbitrary. Choose ô nin). %3D m Determine…

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Q: S* 56sin(t")dt Evaluate the limit lim .8

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Q: find the Value of the limil [z(そ-e 241 lim

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Q: Evaluate the following limits: So ksin(t')dt i). limz »0° =? for any nonzero integer k i). lim, »0…

A:

Q: Compute the following limit. 1-e3 3x as x approaches 0

A: To find,  limx→01-e3x3x

Q: to evaluate the limit. 3 1 v 78. lim x→ x Jo im In(1 + e') di | In(1 + e') dt 2

A: Answer is  1/2 Explanation is given below

Q: To apply the divergence test for ,3/n² n=1 we need to evaluate lim e3ln? n→∞ What is the value of…

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Q: Ql(i) Prove that Lim t* ... /2n²+1 2n²+n]

A: The series is Lim12n2+1+12n2+2+......+12n2+n=12

Q: Show that for any &> 0 and Sn = 1X₁, lim P(|S₁ − 0| ≥ ɛ) = 0. n-00

A: We will use Chebychev's inequality to find the limit of given probability.

Q: Find the limit. 3 - ti + In(t)j k t-3

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Q: (a) ' Compute the following limits 7x3-5x+1 (i). limr→0 4p3+a²+x-8* (ii). lim,0 V5x+1–1 5x (iii).…

A: a) limx→∞7x3-5x+19x3+x2+x-8= limx→∞x37-5x2+1x3x34+1x+1x2-8x3limx→∞7-5x2+1x34+1x+1x2-8x3=…

Q: Prove the following. (b) lim( Vn² +1 – n ") = 0 - (c) lim Vn? n) = } 1 2 ||

A: Prove the following limits

Q: Evaluate the following limits: So ksin(t* )dt i). limz s0 =? for any nonzero integer k t16 ii). lim,…

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Q: Consider the following function. f(x) = 6x2 - 2x Find the limnit. S(r + Ar) – / (r) lim Ar-0 Ar

A: Given , fx=6x2-2xTo evaluate , lim△x→ofx+△x-fx△x

Q: Q3. Compute the following limits if they exist. 1 (d) lim x sin

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Q: Find the following limit. lim t-3 - 5t² – 4t – 1, t2 + 6t – 27 , 4 · 4. cos(t) > < t – 3

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Q: Find the value of the following limits: sinny 5-limy-1 y-1

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Q: Find the following limit. t? + lt – 12 lim t – 3 t-3 answer = <

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Q: п-1 1 The value of the limit lim is equal to - 1+ (2k/n)? n k=0 1 dt 1+ t2 1 dt 1+ t? 2 1 P 1+4 dt 1…

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Q: Solve the following expressiol Vx2 + 3 lim x→1 x² – 1

A: Here we have to solve,

Q: 3. Convert the limit below into a definite integral and then evaluate it using FTC. 3\²3 1+ n)+2(1+…

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Q: 3) see the attached file 3-the limit of this function cos x-(2 cos 2x -1) lim equal to 1-cos x A)2…

A: Given function is ; cosx - (2cos2x - 1) /(1-cosx) at x tendes to zero we find the limit .

Q: Find the odd periodic extension off(x) given that f(x) = x + 1 for 0 < x<2. Write explicitly the…

A:

Q: 1. Show that for all f e L'(R) lim

A: To show: limδ→1∫f(x)-fxδdx=0 Concept used: ∫fxdx=Fx+c Chain rule: ∫fxadx=1aFx+c

Q: Examplesa using the definitin of limait to .2 2. Prove i lim Z =Z

A: In mathematics, a limit is the value that a function (or sequence) approaches as the input (or…

Q: Evaluate the following limits: So ksin(t® )dt i). limz-»0 =? for any nonzero integer k ii). limz-→0…

A: Topic- limit and application of derivatives

Q: Evaluate the limit T: x lim cos 4 x+11

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Q: 1 -1 B lim² =? Limit mevcut değildir D E

A: We will solve the following

Q: Find the lim inf (xn) and lim sup (x₂) of Xn = COS nπ

A: I have calculated the limits

Q: Find the limit, if it exists. (If an answer does not exist, enter DNE.) 1 lim X - co 4x + 3

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Q: Find the limit. TTX lim cos 6.

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Q: Let an for all n e N. Working directly from the Vn+1 definition show that limn- An = 1.

A: To evaluate limn→∞ nn+1: Here, an=nn+1 an can also be written as: an=1n+1n Simplifying this we get,…

Q: Evaluate the limit: limo* [cos(3x)]= O e-3 03 e 01 ◄ Previous

A: To evaluate the limit.

Q: 'e in + = "T 2n Find lim a

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Q: Find the following limit. t2 + 1t - 6 lim - t- 2 answer = <

A: Given limits: limt→2 &lt;4t2+5t-4, t2+1t-6t-2, 2·cost&gt;

Q: Find the following limit. t2 + 5t – 36 t – 4 lim t+4

A: First, we have to find the limit limt→43t2-1t-5. limt→43t2-1t-5=3(4)2-1(4)-5=48-4-5=39 Now, find the…

Q: Find the limit 1- sin z In (100r² + e*) (a) lim (b) lim 1+ cos 2r 100z

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Q: 1. Find the following limit, if it exists. lim (z.1)-(0.0) V2 + y? +1-1

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Q: 3. Prove the following using the epsilon-delta definition of a lir a. lim (2x – 1) = 3 x→2

A: We will prove the limit using definition.

Q: 1. By introducing suitable limiting processes in 1 8(x) = 27 eikr dk, show that (a) 8(x) = 1 1 lim…

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.