1 Consider the function f that is defined over R- {3} by: f (x) =x- 2+ Designate by (C) its representative curve in an orthonormal system 0;i, . 1. Determine limf x Deduce an asymptote to the curve (C). -3 2. Determine lim f x and lim f x 3. Prove that the straight line (d) of equation y =x-2 is an asymptote to (C). 4. Calculate f'(x), then study the sense of variations of f.

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

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Q: 1- Evaluate the limit a-Lim(x/(x-1)-1/ Inx) x-1+ d- lim x+1/√2x²+x+1 x-→∞ b- lim [In(cos(x))]…

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Q: x – 4 8. Prove that lim Vx = 2 using the e-d-definition. Hint: Va – 2 = VT + 2

A: The epsilon-delta definition of limits says that the limit of f(x) at x=c is  value  of limit. if…

Q: lim (Vx + 45 V:

A: Given,          limx→∞x+45-x

Q: 2. Use the ɛ-d definition of the limit of a function to prove that lim = 3. I-»3 T – 2

A: Let  I  be an open interval containing  c , and let  f  be a function defined on  I , except…

Q: 4 cos(20ax) – 4 Let a + 0. Compute the limit lim 25a2 x2

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Q: C. Prove that the limit does not exist. 1. lim (xy)-(0,0) x*+3y

A: We need to prove that limit does not exist.Below find the details solution in image.

Q: Let Sz+8 3x +8 I-1 Approximate lim f(z) numerically and verify your answer graphically. If it does…

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Q: 1- Evaluate the limit a-Lim(x/(x-1) – 1/ Inx) b- lim [In(cos(x))]* c- 2-VX²+4 /sin'x x->1+ x >0 x>0…

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Q: determine the limit  lim_{h->0}[(x+h)2-x^2]/h by treating x as a constant. - determine the…

A: We have to find the limit of the function limx→0cos (2x)x and limx→0sin(2x2)x tan (4x)

Q: Prove rigorously that lim lxl = 0.x→ 0

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Q: tan 3h 3. Estimate the value of lim by creating a table of function values for h h = 0.01, 0.001,…

A: Let y=tan(3h)h Then we make tables for the given values. h y 0.01 3.00090032412 0.001…

Q: Exercise 3.2.8. (Used in Example 4.6.1.] Let ICR be an open interval, let c € I and let f: 1– {c} →…

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Q: 1. Find the limit: lim 1-e-2t 1t, ,1+t² tan t00 '1-t2

A: We need  to find limit.

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: The funciton f in the figure satisfies the lim f(x)=5 as x approaches 4.  Determine the maximum…

A: (A) From the inequality |f(x)-5|<1 , we\'ll find range of f(x) first.

Q: Show that lim e/n = 1 for any constant c> 1. Use the squeeze theorem (or any other method) to…

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Q: 1 Let f(x) = 1, 0sx<2 -x2 + 4x- 3, x22 . Compute lim f(x).

A: Topic -limits

Q: 19 let ɛ > 0 be arbitrary. Determine & as a function of e 2 In formally proving that lim -x + 8 1/4

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Q: 5) a) If a > 0, compute lim a¯*. b) Which approaches its limit faster as x→», y=2* or y=x²?

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Q: Exercise 6. Find the limits. 1 dx. 1+ x2n lim

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Q: 02/Use the l'Hopital's rule to find the limits in t3-4t+15 a) lim t--3 t²-t-12 X-8x² X-00 12x²+5x b)…

A: Given problem:-

Q: (b) Show that lim (x.y)(0,0) 2x6 + y2 does not exist by letting (x, y)→(0, 0) along the curve y =…

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Q: Prove, using the formal definition, that lim→-1 (T+1)² +0o. %3D

A: We have to find the limit limx→-13x+12=+∞.

Q: Q3 :(A): By using " L' Hopital rules "; Find the limits of : 2 (1) lim,-ol; (2) lim 0 x ³ (tan-¹x-x)…

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Q: 6. Figure out the following limits without applying L'Hopital Rule. sin(r- 4) V2r -1-1 b. lim (2²…

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Q: 1. Show that the limit lim (2,y)(0,0) In(1 – a2)+ y* x² + y4 does not exist.

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

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Q: 1. Prove that lim (10r – 7) = 3 using epsilon-delta definition of a limit.

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Q: f'(0) = Is f(1.5)>0? ('Yes on not) lim fCh-)-f(-)

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Q: Prove that lim vx sin (2) = 0 using the Squeeze Theorem. x-0+

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Q: 2,3,4

A: (2) Evaluate the limit as follows.Substitute x = 2 and y = 3 in the given expression.

Q: Prove rigorously that lim sin 1/x does not exist.                                    x → 0

A: Let the given limit be y - y = limx→0 sin1xy = limx→0 sin1x1x×1xy = limx→0 sin1x1x×limx→0 1xy =…

Q: 1 Find lim dt

A: The given data is: limx→∞1x∫1xdtt  To find the value of the limit.

Q: 1. Prove, using the e-6 definition of limit, that: (a) (b) lim lim_2²+1=2 24-1 x³+x²+x+1= 4.

A: Limit

Q: .t 5-1 lim t t → 0

A: By L' hospital rule we solve above limit.

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: Suppose that functions g(t) and h(t) are defined for all values of t and g(0) = h(0) = 0. Can…

A: Given that functions g(t) and h(t) are defined for all values of t and g (0) = h (0) = 0.

Q: Find the maximum value of 8 > 0 that satisfies the limit claim corresponding to ɛ = 0.1, that is,…

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Q: - Find the limits using L-Hopital's Rule: cos 3x 1 (а). im $ x→0 X* (b). lim (e - 1)*

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Q: 3. Compute the limit algebraically Vx +3x2 lim 6x 1

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Q: 18. Compute, from first principles only, without using L'Hopital's rule eh – 1 lim h-0 = 1

A: given is  limh→0  eh-1h=1we know    eh  = (1+h1!+h22!+h33!+.............)limh→0  eh-1h = limh→0…

Q: Q3 :(A): By using " L' Hopital rules "; Find the limits of : 1 (1) limx-o[: 1-cos x 2 sin²x A 1.

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Q: 17. Find limx cos (4) using Squeeze Theorem. x-0

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Q: 1 11 lim X- 1 x2 + 9x – 10 Give your answer in exact form. If the limit does not exist, enter DNE.

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Q: Exercise 3 Prove that t2 lim (1+ (19)

A: The given expression is in the form of one to the power infinity. Let L=limp→∞1+t2pp+12. In the…

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.