Prove the following case of L'Hospital's Rule: let ƒ,g be differentiable on (a, +) for some a ER. Suppose that 1. lim g(x) = +; 2. lim I→+∞ 9'(x) f'(x) = +0. Show that f(x) lim = +0. z→+00 g(x)

Q: Evaluate the limit using L 'Hospital's rule: lim x to 0 e^x -1/sin(12x)

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Q: Verify that L’Hôpital’s Rule applies and evaluate the limit.

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

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Q: Prove that

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A: Explanation of the answer is as follows

Q: Evaluate the following limits. 1 - cosh x 1. lim x+0+ ln(x² + 1)

A: Note:- Since you have asked multiple question As per our policy, we will solve the first question…

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. 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. 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: Express the limit as a definite integral. lim 40 E xk A xk; [-5, 1] k=1 1 9x8 dx -5 x9 dx x9 dx x dx

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Q: Express the limit as an integral. * lim (5(x)2 - 2(x;)³)Ax over [0, 3] i = 1 dx

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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 interval. lim x, In(2 + x,') Ax, [0, 3] n- 00…

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Q: Express the limit as a definite integral. n 7 lim 8 1 + i = 1 n- o n dx

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Q: V5+s-V5 lim s→0 S

A: lims→ 05+s-5s=lims→ 01s+5+5Plug in the value s=0=10+5+5=125

Q: Find the value of the following limits: 5y 1-limy→o tan 10y 2-limx-0 6x cot3x 7x²+8x 3-limx-0 4x²…

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Q: Express the limit as a definite integral on the given interval. lim n - 00 cos(x}) Дх, [2л, 5л] Xi i…

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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: Find the limit as x approaches c of the average rate of change of the following function from c to…

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Q: Express the limit as a definite integral on the given interval. lim [4 – 3(x¡*)² + 6(x¡*)>] Ax, [0,…

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Q: Express the limit as a definite integral on the given interval. lim (2-5(x,)² + 6(x;")³] Ax, [0, 4]…

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Q: Show that the limit xycosy lim (x.y)(0,0) 2x +y² does not exist.

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Q: (4-x-x² if x ≤1 if x > 1 2x 5 Calculate the following limits. Enter "DNE" if the limit does not…

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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: Use a table of values to estimate tne value of the limit of lim Vx + 4 – 2 f(x) f(x) 2.

A: Consider: limx→0x+4-2x   Let fx=x+4-2x Since we have to calculate limit at 0 of fx=x+4-2x.  Make the…

Q: (-cx² + 2x, (sin x + cos x, x> n x<π 9) Prove that lim (x* sin+5) = 5 10) f(x) = What value must c…

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Q: Find the Taylor polynomial of degree 3 for sin⁡(x), for x near 0: P3(x)= Approximate sin(x) with…

A: The taylor polynomial for f(n) at x=a in…

Q: Use L'Hospital's rule to evaluate the limit sin(3t) dt lim X-0 x2 O O

A: We have, limx→0∫0xsin3tdtx2 On solving further the integral part, we get the result as…

Q: Q1) A) Find the tangent equation for the function: f(x) = vx2 + 5 at =2 ? B) Find the limits by…

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Q: Express the limit as a definite integral on the given interval. x; In(1 + x3) Ax, [2, 3] lim 3 dx

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Q: If ƒ' is continuous on [0, ∞) and lim→∞ f(x) = 0, show that I p'(x) dx = = -f(0)

A: First we will integrate then put limit

Q: cosx * sin? (3x) 2 2. lim x→0 x2

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Q: verify that L’Hôpital’s Rule applies and evaluate the limit.

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Q: The improper integral f(x)dx can be written as lim f(x)dx + lim_ f" f(x)dx . w--00 w+00 If f(x) =…

A: use the above definition of improper integral

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 t lim n-oLX; In(1 + x;²) Ax, i=1

A: Given that: 1. limn→∞∑i=1nxiln(1+xi2)∆x, [0, 1]2. ∫13(x-4lnx)dx

Q: e2z – 1 lim sin(x) ,2x I→0

A: We need to use L'Hopital rule to find the limit of given function at x=0

Q: Find the limit analytically: 2-4x lim 40 5+3x 1. x - 2x-15 lim x 5 2. Find dy/dx of: x' (x- y) -x-y…

A: The given data is: The strength of the rectangular beam is proportional to the square of width and…

Q: Find the limit using L'Hospital's Rule. e 2x – 1 lim x→0 sin x

A: We use L hospital's Rule

Q: verify that L’Hôpital’s Rule applies and evaluate the limit.

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Q: Evaluate the limit, using L'Hôpital's Rule if necessary. In(e6t – 1) dt lim X> 00

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Q: S 10 – x – a? if x 1 Calculate the following limits. Enter "DNE" if the limit does not exist. 2x +…

A: The given function is f(x)=10-x-x2if x≤12x+6if x&gt;1 We have to calculate,  a) limx→1-f(x)=?b)…

Q: Estimate the limit numerically or state that the limit does not exist (DNE): sin(9x) lim x →0

A: Please refer the attached image for complete solution. THANK YOU.

Q: Q2: (a) By using " L' Hopital rul (1) limx¬(1/2)-(tan x)cos x

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Q: By using The definilion of the D Lim x (X+1)= o 2) Lin x Cos

A: Given: 1)limx→0x(x2+1)=02)limx→0x cos1x=0

Q: Express the limit as a definite integral on the given interval. lim x, In(2 + x^)Ax, [3, 4] n- 00

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Q: find the limit as x goes to infinity of ln(x+1/x+2)

A: Evaluate the limit

Q: 2. Use the given values to set up a table to evaluate the limits. Round your solutions to sin 3x…

A: limx→0sin3xx