In Exercises 21-24, evaluate the limit by making the substitu- ion z = x²+y² and observing that z→0+ as (x, y)→ (0, 0). sin(x? + y²) x² + y2 1- cos(x² + y²) lim 22. 21. lim (x.y) (0,0) x2 + y? (x, y)→ (0,0) e-1//x²+y² e-1/(x²+y²) 24. lim 23. lim (x,y)→ (0,0) (x,y)→ (0,0) Vx² + y²

Q: Prove that lim 12 √16x +9y¹2 2 6 4x² +3y does not exist by showing that the limits along different…

A: We have to prove that  limx, y→0, 0 16x4+9y124x2+3y6 does not exist.

Q: Use the Squeeze Theorem to evaluate the limit: limx→0  x cos(8/x)=

A: We know that cos always has a value between -1 to 1 So,

Q: 1. Use the Pinching (Squeeze) Theorem to evaluate the following limits: (a) lim (-1)"+cos(3n), (b)…

A: As per our guidelines, we are supposed to answer only 3 sub-parts. That is why I will solve only…

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: x²– y? lim (x, y)→(0, 0) x² + y?

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Q: 2] Show that the limit does not exist by considering the limits as (x, y) → (0, 0) along the…

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

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Q: 6

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Q: Use the Squeeze Theorem to prove that lim (r.y)→(0,0) x +y² 0.

A: Using Squeeze Theorem to prove that lim(x,y)→(0,0)x4yx4+y2

Q: Draw the phase portrait for i=2 -1. If z(0) = 1/2, what are the limits lim z(t)?

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Q: Prove that lim (x,y)(0,0) x² + y² does not exist by considering the limit along the x-axis.

A: Given: lim(x, y)→(0,0)xx2+y2

Q: 1 = 1 z→1-2i z + 2i lim

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Q: Show that limx,y)-(-2,0) g(x, y) does not exist given that (x + 2)³y² (x + 2)6 + 3y4' 0, if (x, y) +…

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Q: Find the limit. Enter DNE if the limit does not exist. 15zy lim (2,9) → (0,0) 22+ y?

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Q: cos 2x - 1 lim 6x -0.1 -0.01 0.001 0 0.001 0.01 0.1

A: We have limx→0cos2x−16xfx=cos2x−16xPut x=-0.1f−0.1=cos2−0.1−16−0.1f−0.1=−1.031017599×10−5Put…

Q: Find the limit lim (a,y) (-45,9) xy cos(x + 5y) (Enter 'DNE' if the limit does not exist)

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Q: find the limit.  18. lim t→0 ((sin 2t/ t )i + e^(-t) j + 4k)

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Q: Find the limit. lim zey cos 2x cos 2x P-(T.0,2)

A: limx, y, z→π, 0, 2ze3ycos2xPlug in the value x, y, z=π, 0, 2=2e3· 0cos2π=2

Q: Let f (x, y) = 2x 2 + 3y 2/xy. Set y = mx and show that the resulting limit depends on m, and…

A: Given: f(x,y)=2x2+3y2xy

Q: 2²+3ry I²+y² (x, y) # (0,0) (x, y) = (0,0) f(x, y): Evaluate the limit, or use a two-path test to…

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Q: (a) Show that the limit xłycos y lim (x.y)(0,0) 2x4 + y2 does not exist.

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Q: sin x 9. lim -0.1 -0.01 -0.001 0 0.001 0.01 0.1

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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: Evaluate using the Squeeze Theorem.

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Q: Compute the following limit along lines of the form y = mx, for m 0, and along the parabola y= x.…

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Q: Find lim(r,y)→(0,0) ry" Cos y if it exists or show that the limit does not exist:

A: Solution :-

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

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Q: Let f(x, y) = xy /(x 2 + y2). Show that f(x, y) approaches zero along the x- and y-axes. Then prove…

A: Concept: The calculus helps in understanding the changes between values that are related by a…

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: In Exercises 19 and 20, show that the limit does not exist by considering the limits as (x, y) →→…

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Q: Find the limits. Enter "DNE" if the limit does not exist. lim cos (x.y)¬(0,0) x' – y' x² + y² 4x lim…

A: Find the limits : Given :            lim(x,y)→(0,0)cosx7-y7x2+y2               =  cos 0-00+0…

Q: Calculate the value of the limit lim(x,y)→(0,0) 22xy/x^2+y^2 as we approach (0,0) along the path…

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Q: (a) Calculate the limit lim (z,y)→(0,0) x³ + y*' or show that it does not exist.

A: Note: - Since you have asked multiple questions, we will solve the first question for you. If you…

Q: Find the limit. Enter DNE if the limit does not exist. lim (2,y) + (0,0) x² + 14y2

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Q: Find the limits in Exercises 9–12. sin 2x cos 0 9. lim 10. lim 0→- 30 х r + sin r 2r + 7 – 5 sin r 2…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: fir + does not exists. Show that the limit lim (z,y)-(0,0) z2 + y2

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Q: Compute the limit (x,y)→(0,0) of 8xy / 2x^2+ 4y^2 along the following paths.   (a)  Along the…

A: Compute the limit limx,y→0,08xy2x2+4y2 along the path (a) Along the y-axis (b) Along the line y=3x…

Q: Find the limits. Enter "DNE" if the limit does not exist. lim (z,4)→(0,0) 5x2 + 6y cos 4x + 3y + 3…

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Q: 11. Show that the limit lim (z.y)→(0,0) r² + y4 does not exists.

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

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Q: Show that the limit lim (r.y)(0,0) In(1 – a²) + yª r² + y does not exist.

A: First replace y=mx and then find the limit if the limit is dependent on m then the limit doesn't…

Q: Show that the limit xłycosy lim (x.y)→(0,0) 2x4 +y² does not exist.

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Q: Find the limit or prove that it does not exist lim (2,y)→(0,0) T° cos x + 2y sin y x2 + 4y2

A: We will evaluate the limit limx, y→0, 0x3cos x+2y3sin yx2+4y2  using polar coordinates.

Q: 4. Use the € & definition of limit to show that lim (2z+3i) = 2 + 5i. z+1+i

A: As per our guideline, we are supposed to solve only first question. Kindly repost other questions as…

Q: 2xy Show that lim does not (x,y)→(0,0) x² + y² exist.

A: The limit function is limx,y→0,02xyx2+y2

Q: 2. Find the limit using Squeeze theorem 1 lim,2|a – 2| sin( `x – 2

A: Given: limx→2x-2sin1x-2

Q: Show that the_limit exists : 2x²siny 2), lim (x,y)→(0,0) 2x² + y²

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Q: Let f(x,y,z)=x^2+4y^2+2z^2. Does the limit lim_(0,0,0)f(x,y,z) exist? If so, use generalized to 3…

A: Given:-             Let f(x,y,z)=x2+4y2+2z2. Does the limit lim_(0,0,0)f(x,y,z) exist? If so, use…

Q: 1. Consider {a} where a, = tanh(Vn). Use our rigorous definition of the limit to prove that lim…

A: As this is a multiple question according to the Bartleby Answering rule, only first question is to…