3.15 let (fa) be a Sequence of function on [₁1] defined by fak+1 (x)= of xe lo1/3] otherwise, رع i if x € [0,1/3] other wise k=0,1,2, show that ----- 1 lim in of the slim inffpak lim sup Passi 0 [2(x) (x)= 20

Q: 1. Use the definition to prove that lim_ (r + 8)² = 25

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Q: If a and b are integers, then a/b is always a rational number. True False

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Q: Evaluate the limit. 1² – 4x + 3 lim 3x3 21 +3

A: To evaluate the limit limx→∞x2-4x+33x3-2x+3

Q: Evaluate the following limits of rational functions: a) lim (11 + I *) = !! %3D 3x 11o b) lim 276 +…

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Q: 2 Compute limx→16; 1-x2 1 X-1

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Q: 3 7) Show that lim x²³=c³ for с 3 3 lim x³ = c²³ let 6 (x) = x³ any CEIR

A: In this question, we will show the limit of the function.

Q: 10. Find each limit if x→∞0 a) lim 3-2x 3x x² + 3x-18 (x-9)² b)lim- 7 x +31 c) lim(5x +4--

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Q: Evaluate the limit using L’Hôpital’s Rule

A: The given limit is limx→∞5x2-3x+13x2-5.

Q: 2.3 Q14

A: The objective of this question is to find the limit of the function f(x) as x approaches 2, given…

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A: We know that: f'(x)=limh→∞f(x+h)-f(x)h and f'(a)=limh→∞f(a+h)-f(a)h.

Q: anx" + ... + ajx + ao_ with a. >0 and bk > 0, show that brak + 5. If f(x) = lim f(x) = +∞ + b1x + bo…

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Q: (6- x if x < 5 1. (9 points) For the piece-wise defined function, f(x) = {2x – 11 if x 2 5 na. a)…

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Q: 2x2 – (3x + 1)Vx + 2 lim x-1 x - 1

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Q: Determine the limit value of each of the following functions 11. lim 5(3x + 2) x →-1 12. lim * x→ 4…

A: Tp evaluate the given limits.

Q: 1 Find lim 3 s-2- \s-2 s²–4,

A: Given, Evaluate lims→2-1s-2-3s2-4

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A: Step 1:Step 2: Step 3: Step 4:

Q: Find lim *2 x -2 In(x- 1) by using the La'Hospital Rule.

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Q: lim х-0о (2х-1)3

A: We need to find the limit of limx→∞x42x-13

Q: Q 4. (a). Show that limit *1, does not exist if (5-4x S(x)=. -)- {ax-2 for x <1] for xz1]

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Q: Let an 2n+4n-3 Using the definition of the limit, prove that lim,∞ an 3n2+2n+1* 2/3.

A: Given that, the rational function  an = 2n2+4n-33n2+2n+1 To prove that limit of function is limn→∞…

Q: Explain why the function is discontinuous at the given number a. F(x) = { ex lim f(x) = X-0- lim…

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Q: Evaluate the limit using L'Hopital's rule

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Q: – 32 x3 + 6x? – 32 lim x--4 x3 + 5x2 + 4x

A: we got 0/0 so we apply L'Hopitals rule Take derivative of top and bottom by applying power rule

Q: 10. Given the functions /(x) = 3x² + 2x + x+1 and g(x) = 2x² – 1, find (f - 9)x. a 3x +x b. 3x +x+2…

A: Since you have posted multiple questions i can do first question as per our company guidelines…

Q: In each part, find limx→a f(x), if it exists, where a is re placed by 0,5+, -5-, -5, 5, -∞, and +∞.…

A: The given functions (a) (b)

Q: 22) Find the limit equation of f'C4) if リメト=(×23 で allim Jx -2 x-4 b) lim X-4 C lim x-4 X-L X-4 a)…

A: Here we will find the limit equation for f'(4),