5. Suppose that limn00 Tn = 3. Prove directly (i.e. NOT by applying limit laws) that limn→o = 3.

Q: Evaluate the limit, using L’Hôpital’s Rule if necessary

A: We first try to find the limit by substituting x=1

Q: (b) Show that if lim sn = +o∞ and lim tm > -x, then lim($n + tn) = +0.

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Q: 4. Find the limit if it exists. Use L’Hospital Rule where appropriate, give a reason. ae* – ae-* -…

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Q: Compute the following limits.  limx->2 1/(x-2)5.

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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: 19. lim (-1)"+' (") does not exist. n00

A: Given  The limit is limn→∞(-1)n+1n-1n

Q: x³+3x 7. Compute the limit limx→∞ −²+2° O∞ 0-∞ Oo 312

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Q: Use the E-8 definition of a limit to prove lim Xy (Xy) →(0,0) x² +y²

A: Consider the equation

Q: In(7)/ In(x) (17 + 8x?) Find lim Limit is equal to (Type an integer.)

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Q: Using e – N description to prove that lim - = 0.

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Q: 2. Let r > 1. Prove that lim r" = 00,

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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: f, there is a sequence (zn) in C - {1¹} such that : lim z, = 1, but f(zn) n n→∞ has no limit in C U…

A: Take sequence z(n) as Z(n) = 1 + {[(-1)^n)]/n}So that as n goes to infinity  Z(n) goes to 1 .

Q: 8. Use L'Hopital rule to determine whether the limit exists, and evaluate it if it does. 1 - t lim…

A: Given,          limt→0et-1-tt2

Q: 1 Use l'Hopital's rule to evaluate lim I+0+ In(1+x)

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Q: If [ f(x)\dx<∞ show that limf(x)dx=0 k→∞00 1

A: We have to prove that given expression.

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

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Q: 8 lim is x0o 1+81/x А. 8 B. no limit C. D. 4 Е. О o| 8

A: Find limit of a function 8/1+81/x when x tends to infinity.