(1//1 + s lim s 0 – 1

Q: Example: Examine the convergence of COs x dx (ii) / x'e*² dx

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

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

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Q: 1) Find the dexivative by the limit process. fex> 2X - 3 3X +4

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Q: Show that if Σ i=1 converges towards a € R , then lim ) a; = 0 i=N

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Q: Suppose x1 : 1/2 and xn+1 := x(n/2). Show that {xn} converges and find lim xn.

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Q: 2 then E " Cn+1 converges. 3.11 limnr+m

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Q: lim b-> -0.1x e 1.

A: Let I=limb→∞∫1be-0.1xdx

Q: nl (n + 1) n" (n + 1)(n+1) nl lim 1, because 0 < < 1. n00 nn 1. OTrue OFalse

A: Solve the following

Q: -3k sin() - cos(;)| for k = 1, 2, ... and let A = lim y00 A(k). Then Let A(k) Select one:

A: We have defined a sequence of matrix as  A(k) =  - 3kSin(1/k) - Cos(1/k) e^( -1/k2 )  e^( -k2 )…

Q: IS In(x² – 1) 6. Evalaute lim x→0 cos(x) - 1

A: To find the limit of the function we have to find right as well as left limit of the function. First…

Q: By whatever justifiable means, find In(1 – 22) lim sinh z e – In(e – z) - and lim z+0 z – 1+cosh z |

A: To evaluate the given limits

Q: Show that for any &> 0 and Sn = 1X₁, lim P(|S₁ − 0| ≥ ɛ) = 0. n-00

A: We will use Chebychev's inequality to find the limit of given probability.

Q: 3. Express the limit as integral: n lim Σcos (27xt Ar 2(2πχ* ) over [0,1]

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Q: Discuss the existence of the Limit Limit Cos 1/x-1 x-1

A: we have to check the limit of the given function. We will use definition of limit

Q: 2 for al 2 0. 39. Prove that lim x* cos %3D -

A: Given,

Q: 1. Find the following limits. e3r lim 3.x е (a) (b) lim e3/(2-a) lim In(1x2 - In(1 x (c) _ (d) lim…

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Q: Why does the limit as x goes to 0 of cos(1/x) not exist? Can use two different sequences:…

A: Solution

Q: Q2: A. Evaluate each of the following limits without using “L’Hôpital’s Rule": (1) limx→0°…

A: 1) limx→0sin(cos(x)-1)x22) limx→-∞5-x3-4x22x3+x2-8

Q: Compute lim 2>0 sin x O 2 O-1 1 undefined

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Q: Suppose x1 : 1/2 and xn+1 := xn2. Show that {xn} converges and find lim xn.

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Q: Problem 7. Suppose that xn 2 0 for all n E N and that lim ((-1)"xn) exists. Show that (xn)…

A: Let lim -1nxn=1 Therefore, For ε&gt;0 ∃ k∈N such that -1nxn-l&lt;ε2 ∀ n≥k ⇒xn+l&lt;ε2 ∀ n≥k, and n…

Q: This ratio lim,0+ f(x) f(0) of the limit, and the initial value f(0) is called Euler's number e. It…

A: As given in the example:

Q: Let fn(x) = n²x²(1 – x) on [0, 1]. Does lim,oo fn(x) exist for every r E [0, 1]?

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Q: ()- In(x) C. lim %3D

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Q: 1. Show that the area under the curve y = x³ from x= 0 to x= 1 is equal to the limit lim li=1 2.…

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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: (i) Evaluate In (1-z³) lim 2->0 sin (2z

A: We need to evaluate the limit If the value of the function after putting the limit is in the…

Q: 2. Calculate the following limits: (a) COS x – 1 lim x40 sin x In(1 + x)

A: To find the limit Lim(x tends to 0) [cosx-1]/sinxln(1+x)

Q: Find lim (e-2x cos x) (Hint: Squeeze Theor

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Q: Suppose lim an=2 Determine sadius and interual terval os os ds converyence the pocwer seares…

A: We need to determine radius and interval of convergence.

Q: ) Evaluete tue limeit: lim (e*+3x) using L'Hopital's Peele if hecessany. af

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Q: 2. The limit of the following function equals: x2/3 + x-1 lim x--00 x2/3 + cos2 x

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Q: (3+4y-y is th e of lim y+ oy +3y+4)

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Q: EXAMPLE 4 Investigate the following limit. lim sin 5n x +0 x

A: In this question we have to investigate the given limit.

Q: Question 11 Evaluate the limit lim (ln(3x² + 2) - In(x² + 1)) O ln 3 O In 3 01 00 O ∞

A: To evaluate the following limit.limx→∞ln3x2+2-lnx2+1

Q: 1 Find lim dt

A: The given data is: limx→∞1x∫1xdtt  To find the value of the limit.

Q: 8. lim In(+h)–1 A. 0 B. - C. 1 e D. e E. Does not exist

A: To solve the limit.

Q: determine the radius and convergence of E n=1 5xn/3n2

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Q: %3D In(1+x) 3.A) lim P-ove tha t マー0

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Q: 1 [3] b) Determine whether dx converges or diverges. Justify your answer 1+x?

A: According to the question, we have to determine whether ∫0∞11+x2 dx converges or diverges. If the…

Q: Is In(x² – 1) - 1) 6. Evalaute lim n cos(x)-1 r lulx2-1) lem lulx²-1)

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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: In(x), 5 s x s 16 f(x) A = lim n- o 2 i = 1

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Q: 1 Prove that lim dx exists and find its value. (1+x?)" 1/2

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Q: arctan (2x 7. A) Compute lim arcsin (3x)

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Q: 4. lim In(cosx) 2

A: Given limit, limx→π2-ln(cos x)

Q: 16) [5] Compute the value of the following limits using L'Hôpital's Rule: Ax -1-4x a) lim x40…

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Q: 1. By introducing suitable limiting processes in 1 8(x) = 27 eikr dk, show that (a) 8(x) = 1 1 lim…

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Q: In (x + h) – In x (c) lim h→0

A: To find the limits  Here variable is h , so assume x as constant. 0/0 form , so use L-Hospital's…

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.