Exercise 6: Use Taylor's theorem to evaluate the following limits (а) lim 0 cos(x)1+ x2 /2 3 sin2(ar)2 sin4(x) 3x tan(x (b) lim

Q: Compute the following limits if they exist. sec r – 1 (a) lim z-40- 23 I – sin r (b) lim 1+0 x - tan…

A: L. Hospital RuleIf limx→a f(x)g(x) = 00 or v6c6 formThen differentiate f(x) & g(x) and take…

Q: Evaluate the limits in Exercise 7 and 8. 7. lim sin t dt 8. lim x cos t dt

A: (7) ∫-xx sin t dt =-cos t-xx=-cos(x)--cos-x=-cos(x)--cosx=-cos x+cos x=0 So we get limx→∞ ∫-xx sin…

Q: Figure out the following limits. (No L'Hopital Rule) 2x sin x-a b. lim a. lim 1-0 sinr - tan a -a…

A: Given

Q: Calculate the limits given below In(In x) (sin x)2 a) lim In x b) lim(x + sin x)tan x c) lim х-0+…

A: Part a,Given,         limx→∞lnlnxlnx

Q: Find the limit of each given transcendental function as x→c a. lim x→0 (3tan²x/2x²) b. lim x→0…

A: According to L' Hospitals rule, if the direct substitution of limit will give an answer in the form…

Q: %3D 39. Prove that lim x*cos

A: To prove the given limit

Q: Use l'Hôpital's Rule to evaluate the limit. COS X lim -X- 3. 3. O A. -3 V3 O B. C. 2 O D. -/2 1.

A: We have to find limit As per guideline we have to solve one

Q: 2 dx In evaluating is same as evaluating lim cos ). Determine the value of f(x) if x=0.91 4 C- 2+

A: It is given that, ∫242xx2-4dxandlimc→2+cos-1f(x)c4 It is known that, ∫1xx2-1dx=sec-1x

Q: Evaluate the limit. lim (1+ sin(6æ))cot (= ) x →0+ ||

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Q: sin(In r) 5. Use the Squeeze Theorem to evaluate lim 1-40+ In (sin z)*

A: Solution is given below:

Q: Hence, determine whether the function is continuous at x = 2x. ) Find the value of k such that the 5…

A:

Q: Evaluate the following limits tan x – sin x sin(x – 2) (a) lim x→0 x(-1+ cos³ 2x) (b) lim x→2 Vx + 3…

A:

Q: 4. Use the sandwich theorem to find the following limits: sin x (a) lim cos x (b) lim

A: a) -1≤sin x≤1⇒-1x≤sin xx≤1x⇒limx→∞-1x≤limx→∞sin xx≤limx→∞1x⇒0≤limx→∞sin xx≤0⇒limx→∞sin xx=0(using…

Q: find the limit.  18. lim t→0 ((sin 2t/ t )i + e^(-t) j + 4k)

A:

Q: 9. Use the squeeze theorem to evaluate lim(x – 1)²ecos (Hint: X-1 -1< cos s s 1, x # 1) ズー1

A: If h(x) &lt;=f(x) &lt;=g(x)

Q: Convert the limit to the indeterminate form 0/0 and then evaluate the limit. lim(x-1)tan(pi(x)/2)…

A:

Q: Exercise 13. Show the following results. 1/sin z 1 + tan x (a) lim I-0 1+tanhr = 1 (b) | lim(2* +3*…

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Q: Evaluate the limit. (). lim ( sin i + cos 3' j + tan(t)k

A: Limit (f1 i + f2 j + f3 k) = Lim f1 i + Lim f2 j + Lim f3 k

Q: The value of limit x→0 of {sin (a + x) – sin (a – x)}/x is

A: we have to find the value of  limx→0sin(a+x)-sin(a-x)x

Q: Find the value of each limit: 1- cos x а. lim x→0 x+x b. lim sin(x+ h) – sin(x) h h→0

A:

Q: Evaluate the limits in Exercise 7 and 8. cos t dt sin 8. lim x x→0+ 7. lim

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: Compute the following limits if they exist. tan x x – sin x (а) lim 0+ Vx (b) lim (c) lim tan x- seC…

A: We have to find the limit

Q: Pr. #3) Prove that the limit does not exist. cos(4ry)- 1 lim (x.v)-(0,0) 2r + 3y4

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Q: c) ) et 2. Aprēķināt robežas sinx-x a) lim X-0 b) lim X0 cos x-1'

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Q: B. Evaluate only three of the following limits tan 30 lim 0-0 sin 80 x+1 i. ii. lim + zX co-x 1 = 2…

A:

Q: Express the limit lim n→∞ a = n b = i=1 Provide a, b and f(x) in the expression (6 cos² (2πx)+5) Ax;…

A: To find value of a, b and f(x).

Q: 1. Prove that lim vx sin O using the Squeeze Theorem.

A: Using squeeze theorem,

Q: Evaluate the limits sin3h-2h 4) lim sin?x 5) lim x 0 1-cos x

A: (4) limh→0sin3h-2hh=sin3(0)-2(0)0=00=indeterminate form Apply L'Hospital rule,…

Q: a(sin 2a+cos a sin 2a) lim 1-cos² a a→0

A:

Q: 3. Use Squeeze Theorem to evaluate lim 1-0+ sin(In r) In(sin æ)"

A:

Q: 2. Calculate the following limits: (a) COS x – 1 lim x40 sin x In(1 + x)

A: To find the limit Lim(x tends to 0) [cosx-1]/sinxln(1+x)

Q: )Use the Sandwich Theorem to show that lim cos x-1 =0. x2 Find f'(x) for f(x) = sec(vx2 + 1).

A:

Q: Using the squeeze theorem, find the value of lim,0 x² sin=) O 1 O -1 O does not exist

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Q: Evaluate the limit. lim (sin(4t)i + cos(5t)j + tan(2t)k)

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Q: d cos x = – sin x - dx

A:

Q: Compute the following limits if they exist. sec x – 1 (а) lim x – sin x (b) lim I-0 x – tan r

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Q: Find the limit. lim vx+11 cos (x + 3T) X-- 3T lim vx+11 cos (x + 3r) = →-3T Type an exact answer,…

A:

Q: Find the limit of tan^-1(e^(x^2)+x^2+1/x) as x approaches negative infinity

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Q: Use the Squeeze Theorem to prove that lim væ[1+ (sin(27/x))²] = 0.

A: "Since you have asked multiple questions in a single request, we will be answering only the 1st…

Q: Evaluate using the Squeeze Theorem.

A: Given: limx→0 x sin 1x2  To evaluate: The limit using Squeeze theorem.

Q: - Evaluate the limit a-Lim(x/(x-1) – 1/ Inx) b- lim [In(cos(x)) c-2-VX-+4 /sin'x x>1 x>0 d- lim x+1…

A: Limit values

Q: Evaluate the limit lim sin(T cos x) 1 C T A,

A: we need to calculate Limx→π sin(πcos(x))Now we will substitute x = π to get answer .

Q: 1. Find the following limits proving whether they exist or do not exist y² sin x X E) F) lim (x, y)…

A:

Q: 70. J, f(x) dx may not equal lim , f(x) dx Show that 2x dx o x² + 1 diverges and hence that 2x dx x²…

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Q: 2 cos(4x) – 4x² – 2 4. lim х-о sin(2x) —х2 — 2х

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Q: Find the limit or prove that it does not exist lim (2,y)→(0,0) T° cos x + 2y sin y x2 + 4y2

A: We will evaluate the limit limx, y→0, 0x3cos x+2y3sin yx2+4y2  using polar coordinates.

Q: Evalvate the limit lim Cos x In(x- TT Xーつク nピ-e)

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Q: Use the Squeeze Theorem to show that lim x* cos () = cos = 0.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 1. Consider {a} where a, = tanh(Vn). Use our rigorous definition of the limit to prove that lim…

A: As this is a multiple question according to the Bartleby Answering rule, only first question is to…