14. Suppose that lim s,-s, with s>0. Prove that there exists Ne N such that s,>0 for all n≥N.

Q: Give an E-S proof for lim 2x²+1=3 x+1

A:

Q: 11) lim [6-x| 11) x -6 6-x A) 1 B) Does not exist C) -1 D) 0

A:

Q: formet Prootice2. finkch proof thae lim nå =1

A: Show that limn→∞ n1n=1

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: lim z+di = 1 Z-1-21 I lim 2 Z-0 lim 240 (Im²) ² 12₁²2² does. (Rez) --121 - compute 0 lim 7731 Z 27²…

A: We will find the required.

Q: +h)4-44 Q 9 Compute limh-0 h

A: To find:

Q: 84. Prove that if lim ar = L, then lim ak11 = L. k00 %3D k00

A: limk→∞ak=L

Q: Prove: 1 a) lim Z+2i Z-1-21 b) lim 2-0 c) lim - 1 (Im²) ² IZE does not exist (Rez) 12-1 = 0

A:

Q: lim (3 x4 x→-0 x3 Select the correct response: -8 O 3 Limit does not exist -2

A:

Q: 9" 45. lim n+00 n!

A: Here given as

Q: 27. Use the definition of limit and the Archimedean Property to prove lim 1 n-xn-7 = 0.

A:

Q: 10. lim( +1-V +1-)

A:

Q: 2) lim f(x) x-0 -3 -2 A) 1 B) » C) does not exist D) -1

A:

Q: Evaluate N εν Σ √19 lim N→∞ N 19 19 19(1) - 2

A: limN→∞1N∑j=1N19-19jN2

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

A:

Q: 4. Compute limz+2 2 2²-4 Z-2

A:

Q: 7x + 6 17) lim V6x2 + 1 X-00

A:

Q: 71. an 1 1 √n² + n² vật thổ v2nt 8 ≤an ≤ 1 √²n²

A:

Q: x² - 4 lim 1-2 +2 O-00 O 4 O 4 O 2

A: Use  a2-b2=(a+b)(a-b)

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

A:

Q: 5. Let x X for n-1,2,3,.... Then lim "n+ n- X n" n A) e B) e2 C) e D) e E) 1

A: First we plug expressions for x_(n+1) and x_n in the given limit.

Q: Show that if lim supn→∞an = L for a finite real number L and > 0, then an> L+ ε for only finitely…

A: The objective is to prove the statement: If limsup an=Ln→∞ for a finite real number L and ε&gt;0,…

Q: 10. Prove that if lim(x,) = x and if x > 0, then there exists a natural number M such that x, > 0…

A:

Q: Evaluate: x - 4 lim x-4x2 - 8x + 16 O Limit does not exist O 4 O1

A:

Q: 1 Assume that lim xn +oo. Prove that lim n-00 n +o0. n-00 k=1

A: Given limn→∞xn=+∞

Q: 24. Evaluate lim (13 + 23 + 33 + + n°). n-00 n*

A: To evaluate the below limit.

Q: 1 Use the definition of limit and the Archimedean Property to prove lim n-xn-7 = : 0.

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

Q: 84x – 105 - lim x→-2 24x 30 O 3.500

A: limx→-284x-10524x-30

Q: 1. てー d Sinlt).losl). 1-(-1)2 ー1 1. Suppore an an dao 十} an and a, =2." Show that 9nt! the Sequence…

A:

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

A:

Q: 5) Prove: b) lim Z30 (Imz) 12₁²2² does not exist

A: