1. Evaluate the following limits by applying the limit theorems (show solutions): 2. 7. x→4 V4x 1 1 3 I Vx - 5 9. lim x-5 x - 25 1 1 10. lim 2 x→4 x-4

Q: 4] Compute the following limits, using the Algebraic Limit Theorem,where (x„) → 0. 1+2x, a) lim…

A: a) lim 1+2xn1+2xn-3x22 Since limit of xn is zero . Using lim cxn=climxn So, lim 2xn=2limxn=20=0…

Q: XY – 2x - lim (x,y)→(0;2) x² + y² – 4y + 4’'

A: Given:

Q: Jse the two-path test to prove that the following limit does not exist. y* - 2x? lim 4 (x.y)-(0,0)…

A: Since tha value of both the limit is different  Hence the limit does not exist.

Q: x³ + x²y+ 2y³ x² + y? ,3 lim (x,y)→(0;0)

A:

Q: Evaluate the limit: lim yr +6z +3 O-V3 b) 00 V3 d) OV19 -V19

A: Solve for limit

Q: 2. Compute the following limits at infinity and give reasons for your answers. 1 a) lim b) lim 10 4x…

A: 2(g) Given limit is : limx→∞6x2-7x+14x3-7 To find the value of the limit.

Q: Compute the following limits and write your answer to the boxes. lim 2x cot lx %3D 7x2 lim x2 28x -…

A:

Q: Find the followings limits:- х — у 1. lim (x,y)¬(3,3) Vx – Jy. 1 2. lim х-0 \xуx + 1 X.

A:

Q: Compute the following limits, using the [4 -(xn) → 0 Algebraic Limit Theorem,where 1+2xn (a…

A: Let (xn) be sequence such that limn→∞xn=0.

Q: Does the following limit exist? Justify your answer. 2.x2 – xy? - lim (x,y)→(0,0) x² + 2y?

A: First , we find the limit along two different paths . *If two different paths have different limit…

Q: Find the following limits (a) Зп+2 0.3 0.7 0.8 0.2 0.6 0.4 lim 0.5 0.5 0.4 0.6 1 0 0 0

A: Due to the time consumption of the question, we will be answering only the first one. If you need an…

Q: Use the Two-Path test to show that the following limit does not exist. (Use y=x2) lim (x,y)→(0,0)…

A: Using Two-Path test we will evaluate.

Q: Evaluate the limit using a definite integral 2 lim n n 1+

A: Let x = i/n Hence, dx = 1/n when i = 1; x = 1/n; As n tends to infinity, 1/n tends to zero. Hence,…

Q: 4] Compute the following limits, using the Algebraic Limit Theorem,where (xn) 0. 1+2x, a) lim…

A: The given limit is: limxn→01+2xn1+2xn-3x22 Here, treat x2 as a constant.

Q: (a) Determine the following limits without using L'Hôpital's rule. V7x° +4x -x² -2 lim x'-2x-3x+1…

A:

Q: I. Evaluate the following using the limit theorems. lim 1. (x,y)→(-1,4) (5x² – 2xy + y²)

A: The aforementioned theorems allow us to simply evaluate 5x2- 2xy + y2, when x= -1 and y = 4.If an…

Q: If the limit lim (x,y)→(0,0) F(x, y) = 0 along the path of the form y = kx for any real number k…

A: Follow next step

Q: What is the limit of f (x) = 1 as x → T?

A: Value of a constant function is always same irrespective of input.

Q: Use the E-8 definition of a limit to prove lim Xy (Xy) →(0,0) x² +y²

A: Consider the equation

Q: 1. Evaluate the following limits (DO NOT use L'Hopital's Rule): V7+ 2x - 3 (a) lim 1-r 42021 +3 (b)…

A: 1. (a) limx→1 7+2x-31-x

Q: Evaluate the following limits using the limit theorems. Show your solutions. 1. lim (4x – 7) X-0 2.…

A:

Q: 9. lim 15 1 + (i/n)6

A:

Q: (b). Compute each of the following limits. (i). lim,4 ". (ii). lim,0 tanz

A: For (i),     limx→ 4x2-x-12x2-4x=limx→ 4(x+3)(x-4)x(x-4)=limx→ 4x+3xPlug in the value x=4.=4+34=74

Q: Use l'Hospital's Rule to calculate the exact value of the limit ) as x→ 0- Az) fX) = e -1 and g(x) =…

A:

Q: b) Use L'Hôpital's Rule to evaluate the limit L. e2x lim X→∞ x2

A: We can solve part in the following steps.

Q: Show whether or not the following limit exists: xy³ lim (x,y)→(0,0) [x2 + y6

A:

Q: Compute the following limit, or verify that it does not exist: xY lim (x,y)→(0,0) 2x² + 3y2

A: Given that limx,y→0,0xy2x2+3y2 fx,y=xy2x2+3y2 (i) Put x=0 and then y=0 in fx,y find the value of f1…

Q: Find the value of the following limits: sinny 5-limy-1 y-1

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Q: x³ – 16 lim z+too x2 + 4

A:

Q: Find a value of the constant k such that the limit exists.

A:

Q: Find the value of the.limit Olim Jx+z - V3 X+2 @lim Vx+6 x3-2 X+2

A: A number M is said to be the limit of a function f(x) at x=a if for any arbitrary chosen positive…

Q: Compute each of the following limits, or show that the limit does not exist. xY (b) lim (x,y)→(0,0)…

A:

Q: c- Show that the following limit does not exist (x² - y²) lim (x,y)-(0,0) x2 + y² 2

A: Given ; lim(x, y) -(0,0)[(x2-y2)/(x2+y2)]2.

Q: Use two different methods to find the limit where m > 0, n > 0, and x > 0

A: The given limit is: limx→∞lnxmlnxn where, m > 0, n > 0, and x > 0 First method: To evaluate…

Q: Q3. Compute the following limits if they exist. 2 – Vx (c) lim x→4 3 – V2x + 1 -

A:

Q: 2. Find the following limits algebraically. Then support your conclusion numerically. x² – 3x – 10…

A:

Q: Compute the following limits, using the Algebraic Limit Theorem,where (xn) → 0. 1+2n a) lim…

A: Consider the given information. (a) Consider the given expression. limxn→01+2xn1+2xn−3x2n Apply the…

Q: Find the followings limits- x - y 1. lim (x,y)¬(3,3) Vx - Jy 1 2. lim x→0 \xVx +1

A: Given: 1. lim(x, y)→(3,3)(x-yx-y) 2.limx→0(1xx+1-1x)

Q: Let f(x, y) : and let L = lim( y)>(0,0) f(x, y). Which of the following 9x2 + 3y2 statements is…

A: we have, fx,y=xy9x2+3y2 For limit to exist, if we take any curve y=fx through 0,0, the limit along…

Q: x² + x 1. lim x++∞ e²x + Inx 2. lim (sech 2x) esch x² +0+x

A:

Q: Evaluate the following limits for h(x) = { 1- 5z 120 (i) lim h(r) (ii) lim h(x) (ii) lim h(z)

A:

Q: 1. Prove the following limits x 4y4 G) lim (x,y) (0,0) (x²+y4)3 = Does not exist

A: Solution is given below:

Q: Evaluate the following limits, 4- 3y2 – y y-2 6 - y - y? 3 - 7x2 + 14x -8 5. lim 12 -1 2. lim t=2 12…

A: As per guidelines we can answer only one question at a time so if you need more answers kindly…

Q: b) Use L'Hôpital's Rule to evaluate the limit L. e2x lim X→ ∞ x²

A:

Q: 9. Compute the following limits, if they exist. Otherwise, show that they do not exist. 4x2 – 16 –…

A:

Q: 1. Prove the following using the € - 8 definition of limit. (c) lim ² = 2i. z→1+i

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Q: Show that the following limit does not exist: (x – y²)2 lim (r,y) (0,0) ² + y4

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Q: Evaluate the following limits, if they exist. (Do not use L'Hospital rule) Vh? + 15 – Vh + 15 Vh +3…

A: follow next step

Q: Evaluate the following limit: tan(4x“ + 4y*) 4x2 – 4y? lim (z,y)→(0,0) 8x4 + 8y4 x – y O 2 1 2 The…

A: Functions of two variables continuous. Using two path test.

Q: Vx+x? a) lim x+00 5x-x2

A: