Find the limits in Exercises 19–22. dx tan-t dt 19. lim b→1", 20. lim VI - x² 21. lim 2n n + 2 n-00 22. lim (e/n + e2/n + . .. + ea-1)/n + en/n)

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: Prove that lim 12 √16x +9y¹2 2 6 4x² +3y does not exist by showing that the limits along different…

A: We have to prove that  limx, y→0, 0 16x4+9y124x2+3y6 does not exist.

Q: Use the Squeeze Theorem to evaluate the limit: limx→0  x cos(8/x)=

A: We know that cos always has a value between -1 to 1 So,

Q: Verify that L’Hôpital’s Rule applies and evaluate the limit.

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Q: 4 Find the limit: lim t+0 t 4 et – 1, (Enter undefined if the limit does not exist.)

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Q: x – 4 8. Prove that lim Vx = 2 using the e-d-definition. Hint: Va – 2 = VT + 2

A: The epsilon-delta definition of limits says that the limit of f(x) at x=c is  value  of limit. if…

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: 4 cos(20ax) – 4 Let a + 0. Compute the limit lim 25a2 x2

A:

Q: Use the two-path test to prove that the following limit does not exist. x+ 2y lim (x.y)→(0,0) *- 2y…

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Q: Draw the phase portrait for i=2 -1. If z(0) = 1/2, what are the limits lim z(t)?

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Q: B) Find the limits by using L.Hopital rule : 7x 1. lim x→0 sin 2x

A: Here we will find out the required value.

Q: Evaluate the following limits (if exist): 1-cos(ax) (c) lim- a+0 (d) lim/Vxr'+x-Vx°-cx), where c is…

A: To evaluate the given limit we are using L- Hospital rule

Q: Express the limit as a definite in 7 lim 1 + (i/n)3 n→ 0 n i = 1 1 7n dx 8

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

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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: Find the limits in Exercise

A: Given,

Q: Find the value of the following limits: 5y 1-limy→o tan 10y 2-limx-0 6x cot3x 7x²+8x 3-limx-0 4x²…

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Q: Q3 :(A): By using " L' Hopital rules "; Find the limits of : 1 (1) limx-o[: 1-cos x 2 sin²x

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Q: Use l'Hôpital's Rule to evaluate the limit. 2 - 4x+2 lim 2. X>co X+5x+11

A: The limit of a function limx→Xfx=Y means that value of  function fx can be as close to Y by making…

Q: Find the limit. lim (cos(t), cos(t), t In(t)) t-0+

A: Given: limt→0+cost, cost, t5lnt

Q: evaluate the limit of xsin(1/x) as x approaches zero using the squeeze theorem.

A:

Q: 1 is equal to cOs x - The value of the limit lim x2

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Q: Find the limit. lim (cos(t), cos(t), t* In(t)) t-0+

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Q: Find the limits in Exercises

A: Given: limx→∞x57+x-3x57+cos x   ……1

Q: 9. lim 15 1 + (i/n)6

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Q: Exercise 6. Find the limits. 1 dx. 1+ x2n lim

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Q: 3x²+2x- 3 3. prove that with the help of the limit definition (with necessary analy si)

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Q: Find the limit: lim as x approaches 0 of (x^2-1)(2-cos x)

A: Use products of limit formula to find the limit as follows.

Q: 6. Figure out the following limits without applying L'Hopital Rule. sin(r- 4) V2r -1-1 b. lim (2²…

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Q: Find the limits in Exercises 9–12. sin 2x cos 0 9. lim 10. lim 0→- 30 х r + sin r 2r + 7 – 5 sin r 2…

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

Q: Q2: (a) By using " L' Hopital rules "; Find the limits of ::- (1) lim,-(n/2)-(tan x)cosx (2) lim,--0…

A: L'Hôpital's rule states that for functions fx and gx which are differentiable on an open interval I…

Q: Estimate the limit numerically. x² lim x+4 O diverges to +o 00 O diverges to -∞

A: Given

Q: Prove that lim vx sin (2) = 0 using the Squeeze Theorem. x-0+

A:

Q: Compute the limit 1 e¯nt lim dt. VE n-00

A: Given: limn→∞∫01e-nttdt

Q: Estimate the limit numerically or state that the limit doesn't exist: lim sec (x) T→1+ vx5–1 (Enter…

A:

Q: Find lim r(t) when 5 In t r(t) = (: 2 tan¬lt, e-2 t 2 tan-t, e 1. limit = (2т, —2, 0)

A: Given    r(t) = 2tan-1t , e-2t , 5 lnttwe need to  find  the limt→∞ r(t)=?

Q: Exercise 6. Find the limits. sin (=)dr lim lim 1+r2n xp- lim n Jo (1+ =)"dz. dr.

A: A function is said to be bounded if it's upper bound and lower bound exist. We will use Lebesgue…

Q: Q2: (a) By using " L' Hopital rul (1) limx¬(1/2)-(tan x)cos x

A:

Q: Calculate the limit limx→0cos(x+a)−cos(x−a)x

A: We have to find the limit: limx→0 cosx+a-cosx-ax This is simple limit so putting x=0 in the…

Q: - Find the limits using L-Hopital's Rule: cos 3x 1 (а). im $ x→0 X* (b). lim (e - 1)*

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Q: Q3 :(A): By using " L' Hopital rules "; Find the limits of : 1 (1) limx-o[: 1-cos x 2 sin²x A 1.

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Q: 2. Calculate the following limits. sin 3x a) lim х-0 2х(х-3) V4+h-2 b) lim h→0 h 5х-4 c) lim X→∞…

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Q: (i) Prove that Lim Vent +1 1 1 + + + /2n²+n] ... V2n2+2

A: Given that, g(x)=limn→∞12n2+1+12n2+2+....+12n2+n from sandwich theoram we can say if f(x)≤g(x)≤h(x),…

Q: Find the limit: lim f(x) as x approaches 0 of Kx2 - tan2 (mx)cos(mx)/x2     (m not =0)

A: Consider

Q: B) Find the limits by using L.Hopital rule : 7x 1. lim 0 sin 2 -16 2. lim 4 (V+5)-3

A: 1. Here limx→07xsin 2x is of 00 form, so by applying L. hospital rule, we get limx→07xsin 2x =…

Q: (b) Use the sandwich theorem to show that limy0x°cos - = 0. 1 In order to apply the sandwich…

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Q: Find the limits

A: Given that limx→∞1x∫0xtan-1tdt . First integrate the interval in the given expression.…

Q: What is the the value of the limit lim,0 Vn (Vn+1- /n)? Show your work.

A: