Assume f is continuous on an interval containing zero anddifferentiable for all x not equal to  0. If limx→0 f'(x) = L, show f'(0) exists and equals L.

Q: Express the limit as a definite integral on the given interval. cos(x¡) Дх, [Зл, бл] Xi lim бл dx…

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Q: Express the limit as a definite integral on the giveninterval.

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Q: Express the limit as a definite integral on the given interval. n lim X; In(2 + x;) Ax, [0, 4] i = 1…

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Q: x? sin(1/r) if r + 0 if x = 0. if x + 0 f (x) = %3D

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Q: Express the limit as a definite integral on the given interval. n lim x, In(2 + x;²) Ax, [0, 3] j=1…

A: Given: The limit limn→∞ ∑i=1nxiln2+xi2∆x on interval 0,3 To determine: The interval notation of…

Q: Express the limit as a definite integral on the given interval. [9(x†)3 – 4xj]Ax, [2,9] lim n- 00 i…

A: Express the limit as a definite integral on the given interval.

Q: Express the limit as a definite integral on the given interval. lim 2x,* + (x;")² Ax, [4, 9] i = 1…

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Q: Express the limit as a definite integral on the given interval. in lim [5(x,")3 – 2x,*]Ax, [2, 7] i…

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Q: Express the limit as a definite integral on the given interval. n lim x; In(1 + x²) Ax, [0, 1] i = 1

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Q: Express the limit as a definite integral on the given interval. n lim x, In(2 + x,-) Ax, [0, 4] i =…

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Q: Express the limit as a definite integral on the given interval. [4(x;")3 – 4x;1Ax, [2, 7] lim i = 1…

A: Solve the following

Q: Express the limit as a definite integral on the given interval. n eši -Ax 4 + X; lim [0, 2] i = 1

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Q: Express the limit as a definite integral. [Hint: Consider f(x) = x*.] lim n- 00 dx

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Q: Express the limit as a definite integral on the given interval. lim (5(x)3 – 7x]Ax, [2, 6] n- 00 = 1…

A: Consider the given limit and interval. limn→∞ ∑i=1n5xi*3−7xi*Δx and interval=2,6 Now, go through the…

Q: Find the equation of the tangent line of f(x) = sqrt(x+1) at x = 0 using the definition of limits.

A: To find: The equation of tangent of line. Given: The function is f(x)=x+1. Concept used: Slope…

Q: Express the limit as a definite integral on the given interval. lim [9(x;")³ – 4x;]Ax, [2, 8] i = 1…

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Q: Express the limit as a definite integral on the given interva lim [2(xf)³ – 5x¡]Ax, [2, 5] n- 00 i =…

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Q: Express the limit as a definite integral on the given interval. lim x, In(2 + x,') Ax, [0, 3] n- 00…

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Q: Express the limit as a definite integral on the given interval. lim X; V 4 + x3 Ax i = 1 [2, 3] n-…

A: Given the limit limn→∞∑i=1nxi4+xi3∆x in the interval [2,3]

Q: Express the limit as a definite integral on the given interval. cos(x) lim Дх, [Зл, 4л] X; i = 1 dx

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Q: Express the limit as a definite integral on the given interval. n X;* lim -Ax, [1, 9] (x;*)2 + 8 i =…

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Q: Express the limit as a definite integral on the given interval. lim X; In(2 + x) Ax, [2, 5] i = 1 '5…

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Q: Express the limit as a definite integral on the given interval. n lim -Дх, [1, 3] (x*)2 + 8 i = 1 dx

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Q: Evaluate the definite integral by the limit definition.

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Q: Express the limit as a definite integral on the given interval. -Дх, +7 [1, 6] lim dx J1

A: The following limit is expressed in the definite integral as shown below

Q: Express the limit as a definite integral on the given interval. n *2 lim 2x;* + (x;") Ax, [1, 6] i =…

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Q: Express the limit as a definite integral on the given interval. lim Ax, [1, 4] (x;)2 + 3 dx Need…

A: Given query is to write the expression as integral.

Q: Express the limit as a definite integral on the given interval. lim Ax [О, 6] 9 + X; i = 1 dx

A: The answer is found in step-2

Q: Express the limit as a definite integral on the given interval. lim x; In(1 + x?) Ax, [0, 1] dx

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Q: Express the limit as a definite integral on the given interval. -Дх, [1,4] (x;)2 + 9 lim i = 1 dx

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Q: Express the limit as a definite integral on the given interval. cos(x;) Дх, [Зл, 4л] Xi lim i = 1 4л…

A: By definition of definite integral as limit of sums is

Q: Express the limit as a definite integral on the given interval. x, In(2 + x) Ax, [0, 2] lim dx

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: Express the limit as a definite integral on the given interval. -Дх, [1, 8] lim х? (x;)? + 8 Ax dx

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Q: Explain, using limits, why f(x) = - is not continuous at x = 0.

A: To Determine: explain using limits why f(x)=1x  is not continuous at x=0. Given: we have a…

Q: Express the limit as a definite integral on the given interval. exi Ax 3 + x; lim [0, 4] dx

A: We need convert summation in to integral

Q: Express the limit as a definite integral on the given interval. n -Ax, (x;")2 + 6 [1, 5] lim i = 1 5…

A: Given query is to express the limit as definiye integral.

Q: Express the limit as a definite integral on the given interval. n exi lim -Ax [0, 9] n→∞ i = 1 6 +…

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Q: Express the limit as a definite integral on the given interval. * n Χρισ lim Σ -ΔΧ, [1, 8] 0-00…

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Q: Express the limit as a definite integral on the given interval. cos(x;) lim n-0 i = 1 Дх, [2л, 5л]…

A: Given limn→∞∑i=1ncos(xi)xi∆xi in the interval [2π,5π]We need to express the limit as a  definite…

Q: 25. Let f(x) be a continuous function. Express + ()] lim п—оо as a definite integral.

A: Definite integral in the form of the limit of the Riemann sum:   Let f(x) is a continuous function…

Q: Express the limit as a definite integral on the given interval. lim x; In(2 + x²') Ax, [0, 2] i = 1…

A: Given, limn→∞∑i=1nxi ln2+xi2∆x And interval 0, 2

Q: Express the limit as a definite integral on the given interval. cos Xi lim Ax, [3π, 5π] 818 5m 3m dx

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Q: Express the limit as a definite integral on the given interval. cos X Дх, [Зя, бя] lim 6x X dx 3x

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Q: Use the definition of the definite integral as a limit of a sum to evaluate (x² – 4x - 2) dx -2

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Q: Express the limit as a definite integral on the given interval. lim (4(x")3 – 3x;]Ax, [2, 6] n- 00 i…

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Q: Express the limit as a definite integral on the given interval. lim [9(x,")³ – 4x†]Ax, [2, 6] n- 00…

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Q: is continuous lim x f(x) = 0 = f is continuous in 0.

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Q: Express the limit as a definite integral on the given interval. n lim x; In(1 + x}) Ax, [0, 1] i = 1…

A: Given: To convert the limit function to integral function, the following expression is used.

Q: Suppose  f is continuous on [0, ∞) and limx→∞ f (x) = 1 . Is itpossible that  ∫ 0∞f (x) dx is…

A: Consider a function f(x) which is continuous on [0, ∞) with limx→∞f(x)=1. The objective is to…

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.