2 – 1 | lim X→-1 elx+1)² – 1 | - 1

Q: If || fn-fl|p. os no prove that lim 11 fn V = || f 11 p. له در

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Q: Find the limit infimum and limit supremum of the following x₂ = Yn where Yn is bounded. Хр n lim…

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Q: c) Prove that lim z' - z +1-i =1- z>l+i z – 2z +2 2

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Q: 2.14 Show anx" + ... a12 + ao lim n an bn .. provided that b, # 0.

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Q: 2+ 1 ** Evaluate lim - X+ oL dt A B none of these 1- e? 1 4

A: Given that,  Now taking the integral term,

Q: 22 -y2 =? lim (z,1) (0,0) +y? O A. 0 O B. 1 OC. -1 D. Does not exist Reset Selection

A: Explanation of the answer is as follows

Q: Find the limits

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Q: Evaluate the following limits.

A: Consider the given expression: limx→1xlnx+lnx-2x+2x2ln3x Check the form of indeterminacy:…

Q: Finding Limits #23 please

A: Given information:

Q: 6. Evaluate the limits. sin(0-4) а. lim Ө-0 Ө-4 4w b. im woo 20w+1

A: It is required to calculate the limits of the following expressions:For part (a):

Q: 1. Solve the following limits. (a) lim 1-- V1+ r2 x Inr (b) lim 2-0 I+ In r 2 + sec (3r) (c) lim…

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Q: 1- Evaluate the limit a-Limx²-4/(x²-x-2) X-2 b- lim [In(cos(x))]* c-2-(X²+4) ¹/2²/sin²x x-0 x →0

A: L'Hospital rule:- If a function is of the form p(x)q(x)   then if limx→a p(x)q(x)  is of 00 or ∞∞…

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: Pr. #3) Prove that the limit does not exist. cos(4ry)- 1 lim (x.v)-(0,0) 2r + 3y4

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Q: Show that Hint: Use to obtain In[|x|] √√2/T. 1 - cos(sx) x² 8 1 - cos" x x² lim n-√√n 00 2 π dx |s|…

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Q: Let fn(x) n²x²(1 – x) on [0, 1]. Let e > 0. Does lim,0 m ({x € [0, 1]|fn(x) > e}) exist?

A: Given that fn(x)=n2x2(1-x) on the interval 0,1 The objective is to show whether…

Q: 4x In z (a) Evaluate lim エ→1( (1- 2)-- -1) C.

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

Q: 4-IS. 2-x s Mx) s 2Cos(x) for all x then find: lim f(x)

A: It is an application of sandwich principle of limit

Q: Prove the following. (b) lim( Vn² +1 – n ") = 0 - (c) lim Vn? n) = } 1 2 ||

A: Prove the following limits

Q: In x lim 4!n e %3D In x -

A: Our is to evaluate the limit using L' Hopital's rule given below:-

Q: 14. Evaluate the following limits. (3-x+x2, 5x+1Y e) lim 2x y-0+ 5x-7 lim 2 f) X0o ( 3x2 x sin x lim…

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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: 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: Prove lim(Vn+1- n)=c 8. 0 =

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Q: 10.2.29 Because 0.5 1. e 3+(1/3"-1)

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Q: #1. lim t-tvt 213/2 12 +3t – 5 t→ 00 #2. lim x → - o0 x' +1 #3. lim tan1 (In x) x → 0+

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Q: 10. a.) Use: lim 2(x, )ar = [ Ax)dx , i-1 b- a Ax = and x, = a +i Ax %3D %3D to evaluate (5x² – 2x³…

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Q: (b) lim (r + 1) In x

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Q: B. lim, 0+(1 + 2r)In(2=)

A: Given: limx→0+1+2xln2x   As we know from the limit rule; If the limit is limx→afxgx and it result of…

Q: Determine the indeterminate forms of the following and solve them using L'hopitals rule. a. lim -1-x…

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Q: Examplesa using the definitin of limait to .2 2. Prove i lim Z =Z

A: In mathematics, a limit is the value that a function (or sequence) approaches as the input (or…

Q: -1 lim,→-1 f(x) = O -1 O 2 -2 O Does not exist

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

A: To solve the limit.

Q: 1. Evaluate the following limits: x2 V2x2 + 1 Vx2 + 5 (a) (3 marks) lim (b) (4 marks) lim a→3 2* +…

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Q: 1-Vx+1 lim X →0 x+9 – 3 -3 -6 6. 3 O does not exists

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Q: lim (2,1) (0,0) (1 + (x² + y²) In(2² + y²))

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Q: 2. Let x E (0, 1), use e – d to prove V4 +x – 2 lim 1 4

A: 2. Given x∈0,1, limx→04+x-2x=14                       ... (1) Use ε-δ definition to prove above 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: lim (1,4) (0,0) x2 + y2 Prove that

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

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Q: 2 39. Prove that lim x"cos = O. X-0 X

A: The given limit to be proved is

Q: 3z+ 4y lim (2,0) +(0,0) 2z - 5y OA- OB OCo O D. Does not exist

A: Explained below

Q: ni eoulav sd no 2x + 5 13. lim x→-2+\x + 1八 +x ber

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Q: ー4 1. lim もラー2 t+t+ 1 to

A: limt→-2t2-4t2+5t+1 Plug x =-2 in it (-2)2-4(-2)2+5(-2)+1 Simplifying we get 0/5 = 0

Q: Evaluate the limit. 1- r + In (x) lim I+1 1+ cos (3 7 x)

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Q: 1-e X x2 +1 X By using the limit comparison test, the integral

A: Using, limit comparison test

Q: Express the limit lim > (5 sin(27x; ) + 8)Ax; over [1, 5] as an integral. n→ 00 i=1 Provide a, b and…

A: To find the solution

Q: I - 4 Prove that lim vI = 2 using the e-d-definition. Hint: Va - 2 = VT +2

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Q: 2n2 + n (a) Using the definition (i.e. N, ), prove that lim 4n2 2 112

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Q: 2. If (xn) and (Yn) are sequences of real numbers such that lim (xn) + 0 and lim (xn Yn) exists,…

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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.