2. Compute lim x" (1 + x)* – e). 1/x x→0

Q: Let n be a positive integer. For which n are the two infinite one-sided limits limx→0±1/xn^n equal?

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Q: 5. If b > 0, prove that lim = 0. %3D 1+nb

A: as per our guidelines we are supposed to answer?️ only one question. Kindly repost other question as…

Q: Estimate the limit numerically. ++7,000x³+ 800,000 lim 2x° + 1,269x³ 8. O to∞ 2.

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Q: 9. Evaluate the limit. lim (1+ sin 3x)1/x x→0+

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Q: Compute the indicated limits. 2x + 5 lim cos 3x? + 2x – 5

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Q: Use definiction to १r० v of Conkegene to lim n+4

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Q: In (1 +) lim t1/2

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Q: Enter a whole or decimal number, or DNE for does not exist or undefined. lx-이 4 + | lim x →0-

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Q: Find the value of the limit: 3x -9x e -e lim In(1+x)

A: Using limit to find the value of given function

Q: 2. Use L'Hospital's rule to evaluate the following limit. cosh(3x) – 1 lim x→0 x2

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Q: Compute, from first principles, without using L'Hopital's rule eh – 1 lim = 1 h-0 h

A: (18) We have to evaluate the limh→0eh−1h. We know that the expansion of eh, which is given by…

Q: Compute, from first principles only, without using L'Hopital's rule eh - 1 lim = 1

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

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Q: 1. Show using sequences of functions that lim -nz? e dx 0.

A: Solution is below

Q: Find the limit. lim x-2/In x X x-0+ O -2 O e?

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Q: evaluate the limit  limit as x approaches infinity of x+cos(x)/ x+1

A: To find,

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

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Q: Find the required limit or indicate that it does not exist. lim o+ (in (t6), 12 in t O 6i O Does not…

A: We have to find out

Q: Compute the limit as x goes to 0 of [tan(x) - sin(3x)] / (x^2)

A: limx→0[tan(x)-sin(3x)]x2=[tan(0)-sin(3(0))]02=00

Q: 12. Evaluate the limit ,kT. lim -> sin(-) n→∞ N n k=1 by expressing it as an integral.

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Q: Prove the limit using epsilon/ delta.

A: To prove the statement by using epsilon, delta definition of a limit is,

Q: In(x² – 5) - lim In(x) cos(1/x) lim e6x – e-6x %3D II

A: Given, limx→∞ln(x2-5)ln(x)cos1x limx→∞e5xe6x-e-6x

Q: 2. Compute xe" da using a limit.

A: The given which we have to find the integration from 0 to ♾️  f(x) = xex

Q: 2) If x3 s f(x) sx for x in [-1,1), find lim f(x) if it exists. x-0 A) 0 B) Does not exist C) -1 D)…

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Q: 1. Prove rigorously that the following limit does not exist lim arctan x-0

A: limx→af(x)≠L: There exists ε>0 such that for every δ>0, there exists x such that…

Q: 5. Use L'Hopital's rule to evaluate e* +2 limy-o ex +

A: limx→∞ex+2xex+5x

Q: lim (1+e9)°

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Q: 12. Evaluate the limit n Σ lim -> sin(- n→∞ N k=1 by expressing it as an integral.

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Q: compute limit as x goes to 0 of (sin(x)-x)/(x^3)

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Q: compute lim as x goes to infinity of ln(x)^1/ln(x)

A: Consider the equation

Q: Find the limit: lim cos(5log6(7^x)) x-> negative infinity

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Q: 18. Find (if exists) * lim (x,y)→(0,0) (4 + (x² + y² ) In(x² + y² )) O 1 O Dose not exist - O 4

A: Limit of the two variables function : Let f be a function of two variables .The limit of f(x,y)…

Q: lim el -1 dt 61-12-e

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Q: In(1+ x?) lim x-0 x3

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Q: 10. Prove that lim x2 = 4 using the epsilon-delta definition of a limit. %3D X-2

A: limx→2x2=4

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

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

A: Given: limn→∞∫01e-nttdt

Q: 4 Using the e – 8 definition of limit, prove that lim x→3 3x 1 1 2

A: To prove the limit of given function using the epsilon delta definition of limit which states that…

Q: x²-4x lim |x| (i)

A: Here we calculate the limit and here x is non positive because x tends to 0 negative.

Q: 12 Show that the following limits do not exist. 1 lim X0x (a) (x > 0), 4,

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Q: 18. Compute, from first principles only, without using L'Hopital's rule eh – 1 lim h-0 = 1

A: given is  limh→0  eh-1h=1we know    eh  = (1+h1!+h22!+h33!+.............)limh→0  eh-1h = limh→0…

Q: Evaluate lim S", r³dx . O 4 Does not exist.

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

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Q: Calculate the limiting value for 1 cos(x*) – 1+5x³ lim x→0 х16

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Q: Express the limit as an integral. Š (5(x; )² – 3(x; )³)ax over [0, 2] lim

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Q: 4. lim In(cosx) 2

A: Given limit, limx→π2-ln(cos x)

Q: 1. Use L'Hopitals's rule to find lim c In(e" – 1). |

A: limx→0+ xlnex-1

Q: 5x2 + 10х + 1 :0. lim x4 + x? + 1

A: