Find the limits in Exercises 51–66. 51. lim x'/(1-x) x-1+ 52. lim x'/x-1) x→1+ 53. lim (In x)'/x 54. lim (In x)/x-e) х—е 55. lim x1/In x x→0* 56. lim x'/In x 57. lim (1 + 2x)'/(2 In x) 58. lim (e* + x)'/x 60. lim ( 1 + x→0+ 59. lim x* x→0* 1/x x² + 1 61. lim 62. lim x0x + 2 lim x(ln x)² x→0* 63. lim x² In x 64. x→0* 66. lim sin x • In x x→0+ 65. lim x tan x→0+

Q: 1 Find lim +3 using limit rules. 2.

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Q: Find the limit or determine that the limit does not exist limx→0 (1/(x√1+x) - 1/x) 1+ x is all…

A: The given function is, Simplify and rationalize the numerator of  the given function as follows.

Q: Formally prove that lim 2x+3=7 using the ɛ,8 definition of a limit. x→2

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Q: Evaluate the limit, using L 'Hopital 's Rule where it applies.

A: To evaluate the limit using L-hospital Rule. limx→9x12+x-6x32-27 Solution: It is 00 form. We know…

Q: 4) Find the value of a for which the limit lim x→0 evaluate the limit. sin(ax)-sinx-x x3 is finite…

A: To Find: The value of a for which the limit limx→0sin(ax)-sin(x)-xx3 is finite and then evaluate the…

Q: et +ex -x²-2 Lim- x-0 sin²x-x²

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Q: 2) By using E-S definition of limit, show lim (x+y+xy)cos÷ cos 음-0 COS (xy)(0,0)

A: Let f(x)=x+y+xy cos1x cos1y

Q: Evaluate the limit. limx→ 0+ tan-1(1/x)

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Q: Use l’Hôpital’s rule to find the limits in Exercises . x² . 8. lim x--5 x + 5 х — 2 7. lim x→2x2 25…

A: We’ll answer the 7th question since we answer only one question at a time. Please submit a new…

Q: prove limx->a 1/x = 1/a with epsilon-delta definition

A: We need to prove the given problem using the epsilon-delta definition.

Q: lim(8x + 16) = 24 Use the epsilon-delta definition of limit to prove that æ→1

A: Let's find.

Q: 2. Explain why f(x, y) must approach a unique number L as (x, y) approaches (a, b) along all paths…

A: A number L is said to be the limit of a function f(x,y) as (x,y)→a,b , if for any real number ∈>0…

Q: 2) For lim (5x)4*, do not evaluate! x→0+ a) Which indeterminant form is the expression? b) Set up…

A: a) The given limit expression  =limx→0+5x4x=00 form

Q: Q2: (a) By using " L' Hopital rules "; Find the limits of ::- (1) limx¬(n/2)-(tan x)cos x (2) limx-o…

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Q: Prove the limit. Using the precise definition of a limit. lim x approaches 5 (x2-x-6).

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Q: Evaluate the limit using L'Hopital's rule. limx->0(e^x-x-1)/2x^2

A: Here we solve the limit using L'Hospital Rule.

Q: 2 lim x→1- x²

A: The limit of the function f(x) is obtained as L=limx→a-f(x)L'=limx→a+f(x) Then the limit of the…

Q: 1- 2x Prove that lim = -1 using the e, d definition of a limit. 5

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Q: Determine the infinite limit. X 1 lim x4+ X - 4 -CO

A: We will plug x=4^+ in the expression first. We got 0 in denominator. So lets check by plugging…

Q: lim x" x→0+

A:

Q: & Prove that lim I = 2 using the e-d-definition. Hint: Vr %3D

A: Given: To explain the given statement as follows, Here, prove that limx→4 x = 2.

Q: 4. Definition of limit. How calculate limit, when uncertainty is(1º)? Calculate lim(2x - 1).

A:

Q: Use the e-8 definition of a limit at infinity to prove that lim = 0. %3D

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Q: Calculate the limit limx→1sin(x−1)x4−1.

A: value of sin(x) at x=0 is: sin(0) = 0

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

A: limx→2x2=4

Q: Evaluate the limit - 1 lim x→1 r2 - 1 If the limit does not exist enter DNE. Limit =

A: Given: limx→1x3-1x2-1

Q: B) Find the limits by using L.Hopital rule: 7x 1. lim- X0 sin 2x -16 2. lim X-4 (Vx+5)-3

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Q: 4 Using the e- 8 definition of limit, prove that lim I-3 3r 1 2 -

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

Q: Evaluate the limits algebraically (using limit laws): Vt2 + 9 – 3 lim- t-0 х2 — 1 lim(2x² – 3x + 4),…

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Q: Determine the infinite limit. lim 3 sec(x) x- (T/2)+ X 00 - 00

A: Given limx→π2+3xsecx, to find the limit

Q: 9 -x2 if x 1 .What value of c makes g(x) continuous at x = 1? Show all conclusions. 2 Vx-4 Using…

A: For a function to be continuos, the left hand limit should be equal to the right hand limit.…

Q: Prove the limit. Using the precise definition of a limit. lim x approaches 0 xsinx(1/x)=0

A: To prove:

Q: 1b. Use L'Hopital's Rule to find the Limit:    lim      2tanx/sec2x x->pi/2

A: To find the limit

Q: 2. Find the limit using Squeeze theorem 1 lim-2|x – 2| sin( r - 2

A: Squeeze Theorem- Let f, g and h be functions defined on an interval I. Let c∈I be its limit point.…

Q: Determine the infinite limit. lim In(sin x) X-0"

A: The given function is limx→0+lnsinx. We have to calculate the limit of the given function.

Q: Use l’Hôpital’s rule to find the limit limx→∞(ln 2x - ln (x + 1))

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Q: Q3 :(A): By using "L' Hopital rules "; Find the limits of : (2) limx-o x-³ (tan-¹x - x)

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Q: Evaluate the limit using techniques from Chapters 2 and 4 and using L'Hôpital's Rule. sin(2x) lim 5x…

A: Limit: a)    x→0limsin (2x)5x=x→0limsin (2x)2x*52=25x→0limsin(2x)(2x)=25.1=25We know ,…

Q: Estimate the limit numerically or state that the limit doesn't exist: 2²–5x-14 lim 1²-6x-7 (Enter…

A: find the limit of the function.

Q: 9. Find the indicated limit. lim (6x - 21)³ x-4 = Show any work necessary

A: Solution: As per given data   lim( x tens to 4){ (6x - 21)³} As per simple rule of limit go ahead…

Q: Prove that lim (4x² – 25) = -9. X→2

A: We will prove this by definition of limit as follows

Q: 5. Using the appropriate precise definition of limit, prove that limx-→4+ +0o. Show all work. X-4

A: Here, we need to show the limit by the precise definition of limit. We are given that…

Q: Prove the limit. Using the precise definition of a limit. lim x approaches 2 x3 =8.

A:

Q: Prove using the def. Of a limit (b) limx→0(2x^2) − 3)) = −3

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Q: 7. Show that the limit does not exist by approaching (0,0) lim .2 ,2 + y- (x, y) (0,0)

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Q: Prove using the definition of a limit. lim x→2^− (1/((x^2) − 4)= +∞)

A: Prove that   limx→2-1x2-4=∞

Q: Prove lim(3x–10) =-4 using the ɛ, d definition of a limit. 2

A: Recall: ε -δ Definition :Let f be a function defined in a deleted neighbourhood of 'a' . Then a real…

Q: 1. Use the limit laws to evaluate lim (2x-1)Vx+3 In each step, indicate the limit law applied.

A: Hello! thank you for the question but as per our honor code, we'll able to solve only the first…