12. For use with the Borel Zero-One Law, the following is useful: Suppose we have two non-negative sequences {an) and (bn) satisfying an ~ bn as n→ ∞o; that is, Show an lim n→∞ bn - 1. Σan <∞o iff Σbn < 0. n n

Q: Show limn→∞(1 +1/2+1/3+···+1/n−lnn) exists

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Q: Let a, b E R with 0 < a < b and let n = Van + bn. Prove that (n) converges and determine lim (x₂).

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Q: Find the limit. x³ + 4x lim X→∞ 6x³ - x² + 4

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Q: Let lim f(x)=4. Use the limit rules to find the limit given below. lim 20 X-5 lim 200-

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Q: Find the limit, if it exists. (If an answer does not exist, enter DNE.) lim -0+2 x

A: We evaluate the limit by substitution

Q: 00 Let a, = 1- /1 -. Show that lim an = 0 and that > an diverges. Hint: Show that a, > 2n n=1

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Q: 7. Show that 1 (1+x)(1+x²)... (1+x²") = for all x € R \ {1} and n € N. Then deduce the limit 2n+1 X²…

A: We use the mathematical induction principle to prove the given relation on n.      For n = 0,…

Q: (5) Find the limit lim n→∞ n k=1 k n²+k².

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Q: lím ln(1 - 2n³)¹/n³ n→0

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Q: 4. Let an be recursively defined by ao = 2, an+1 = √5an for n ≥ 0. Either show that an converges and…

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Q: If lim,→2 f(x) = 4 and lim, +2 g(x) = 7.Use %3D these facts and limit laws to evaluate lim, 2[3f(x)…

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Q: Let limf(x)=L(Lbeanyrealnumber) andnbe apositive integer. X→C Whichof the givenstatementsis/are…

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Q: 2 2 - Let S₁ = 3x₁² x₁ + 3x₂² Ax₂ +...+ 3x² Ax, where the Ax;, fill out the x-interval from x = -2…

A: Given that Sn=3x12∆x1+3x22∆x2+...+3xn2∆xn, where the ∆xi fill out the x-interval from x=-2 to x=6…

Q: 3. Show that if then lim an n→∞ = lim a3n+2 nx L = L =

A: We can write n = (3n+2) - 1 - 2n/3. As n approaches infinity, 2n/3 approaches infinity as well.…

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: 5. Find n² n² + 1 lim arctan( 81x

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Q: 1.001 -1.01 101 -1.1 41.51 3 steuleve, ils 101 $+$-2 (s) 2 limit and justify each step by indicating…

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Q: 1. Use an e-proof to justify each of the following. (-1)"=0. (a) limn→∞ n (b) limn→∞ = 0. n (c) lim2…

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Q: 5) Compute the following limit if it exists 2 (x+y)³² - (x-g) ty lim (x,y)=(0,0) (012)

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Q: Prove that 1 (c) 2F₁ (a, b; c; z)

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Q: 15. Find the following limit V2x+5–1 lim x→-2 x+2

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Q: Express the limit lim n→∞ Provide a, b and f(x) in the expression a = (³ (x ) ³ - 4(x²) ¹) Ax; over…

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Q: Calculate the lim sup and lin inf of the following: √1+n² 2n-5 1. Xn 2. Xn where yn is bounded 3.…

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Q: calculus class

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

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Q: Calculate the following limits (a) limn→o ΣL1 (주5 유 + 옮) i ;=1 n5 n2 (b) limn-> (Vi + V+ V++ ) 2

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Q: Use the fact that e = lim (1+ )" and the formula n00 3. 1-() () + – n n+1 to compute the limit of…

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Q: Evaluate the following limit. lim (ln(t) + In(6t + 5) — ln(6t² – 5t+3)) t→∞0 Number

A: Given data

Q: Find the limit. lim √√b(b-7) b→-∞

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Q: . Calculate Lim п? +п —п by using your calculus and algebra skills n→0

A: Definition and methods help to find the required limit of the given function. A value is said to be…

Q: 16. lim (2-3x) = −4 x 2

A: Graph is presented below. As you can see we take e=1 and find d=1/3. The shaded region in green is…

Q: Suppose that Pk k→∞ ||P|| Calculate lim 18 Pk = √(2)(1.5)4 2 +3(-0.5)k H You can use an…

A: Given,        pk=2 1.5k12+3 -0.5k18-17.