2. Assume that lim så and lim të exist, and tn ≥ sn for all n. Prove that limtë ≥ lim sɲ. (hint: discuss by cases that lim të is −∞ or a real number or ∞).

Q: 1) a)lim n(a (aa – 1) =?

A: Given: limn→∞na1n-1 We can write, limn→∞na1n-1=limn→∞ a1n-11n We get the indeterminate form

Q: b For any be R, use definition to prove that lim-=0. n

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Q: Let a,=l.cayider the sequeue fany generated It anti= an (a) Shaw that Lim an= Ø ,where Ø= !t v5 x…

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Q: Find а) b) a? - an ww. lim lim x→a I – a

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Q: i. Using Arithmetic of Limits, find limn→∞ 2n / (3n + 5)&nbsp. ii. Working directly from the…

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Q: Using Arithmetic of Limits, find limn→∞ 2n / (3n + 5)&nbsp. ii. Working directly from the definition…

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Q: Suppose that lim an = 5 and lim bn = 9. n-0 n→∞ Determine the limits. lim (an + bn) = n→∞o lim a =…

A: We are given that limn→∞an=5 and limn→∞bn=9.We need to find the given limits.

Q: Show that the following: If r > 1, then lim Vr – 1 = 0.

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Q: Assume lim,→a f (x) = L. If f(x) > k, prove that L > k. Prove by con- tradiction by assuming L < k…

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Q: sec x lim sec(3x) X- 2 5х lim In(2+e*) х>о

A: Given:        limx→π2-secxsec3x we know that secx=1cosx         =limx→π2-cos3xcosx

Q: 91. Prove the second part of Theorem 1.31(a): If k > 0, then lim x-* = 0. x00

A: Write x-k as 1xk to get limx→∞1xk. Plugin the limit to get 1∞k.  limx→∞x-k=limx→∞1xk=1∞k

Q: 1-cos 2x lim 4x

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Q: O 2x-12 fix)<x? for OLx3, Find lim flIN) <ーフ」

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Q: 1.Solve the following polynomial equations.  a)  2X3+1=x2+2X

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Q: lim 0→0 sin 26

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Q: Determine the limit 2x z 3 lim,> 3z2 1 8x |5 O x=2/3 O x=-5/2 O x=-2/3 O None of the Above

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Q: Q4) Prove each of the following: V12-- 1- Prove that lim -7 3 VO-3 2-If y - (tan3x)" prove that -…

A: Given: 1) limx→3(12-x-x6+x-3)=-7 2) y=(tan3x)2x, dydx=(tan3x)2x(6xsec23xtan 3x+2ln(tan3x))…

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

A: Given Data: Let us consider the given data, limn→∞ 3n2+12n2+2n -1 To Find the following limits.

Q: Evaluate the following linits ) lim lim cos mn Cos nx

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Q: Let

A: We use definition to prove our result.

Q: 1+ V5 Let s1 = 1 and for n > 1, let sn+1 = Vsn +1. Show that lim s, = 2 n00

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Q: Show that the limno (1 + )" exists in a detailed proof, and find the limit.

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Q: Show that both R and R2 may be covered by countably many open balls.

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Q: Assume that lim sn = s and sn>0. Prove that s > 0.

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Q: Find: L . -im Z÷1+1 L Z²+1 2 Z²_1 im Z10 +1 7/0 14-Z 6 Z+1

A: (a). The given limit is: limz→1+iz2+1z2-1

Q: Compute the exist i. lim 8+iz 2--2i z+4 following limies if they . ¡i. Gm Rez.

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Q: 8. Find the following limit. t+1 a) lim < t- +o 3t2 + 2 b) lim (ti- 3j + VE k) t 2

A: The solution is given as

Q: Let un > 0, Vn € N. Also let lim un = μ. Prove that lim (n + 1) Un+1 = μ. n4x 84x

A: