, 36-6x2-y2 2. Evaluate the limit using the properties of the limit: 7xy2 lim (x,y)(2,3) x2 y2 3. Verify the limit using the definition: 7xy2 lim (xy)-(0,0) x + y2 4. Discuss the continuity for: Зх-у a) g(x,y)x2+y2-9 5 b) h(x,y)- y-x4 5. Find the partial derivatives for: a) f(x,y)-4y+xy2 - 5 xy3 b) f(x,y)-e*cos (xy2) 6. Find the slope in the x-direction and in the y-direction of the surface y2 1 f(x,y)-x2++at the point (0,-1,) using partial derivatives. 4 4

Q: Use the definition of the limit to prove that lim (r+2y– 1) = 1. (7,4) (0,1)

A: Given function is fx,y=x+2y-1

Q: Calculate the limit and show proof if it does not exist. lim (r,9)(0,0) (x2 + 2y6)2

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Q: x²– y? lim (x, y)→(0, 0) x² + y?

A:

Q: 3. Use the epsilon - delta definition to prove that the limit is lim Vx x-0 371 when

A: To use epsilon delta definition to show that the below limit is 0. limx→0x3=0

Q: Use the e – 8 defintion of a limit to show that lim (6z – 4) = 2+ 6i z→1+i

A: The given problem is to show and evaluate the given complex limit by using the epsilon-delta…

Q: Let f (r, y) = 5z'y 7+2y0 %3D Evaluate lim)(0,0) f (x, y). Either find the limit or show that the…

A:

Q: evaluate the limit using continuity lim(x,y)→(1,2)(x2 + y)

A: Evaluate the limit using continuity    lim  x,y→1,2x2+y

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: evaluate the limit using continuity. (x² + y) lim (х,у)->(1,2)

A: The statement of continuity states that if f(x) is continuous, then f(a)=limx→af(x), where a is any…

Q: Use the definition of limit to prove lim (r+ 4y²) = 6. (2,y) (2,1)

A:

Q: Investigate whether the limit exists lim (r,y)→(0,0) x²y² + (x – y)²

A: Given :                           lim(x,y)→(0,0) x2y2x2y2+(x-y)2  To find :…

Q: 4. Does the following limit exist? ry?-1 lim (zu)(1,1) y -1

A:

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

A:

Q: Prove, using the e-8 definition of limit, that: (a) (b) lim ²+1=2 24-1

A:

Q: 4-ху Find lim (x,y)¬(2,1) x²+3y2 if it exists, or show that the limit does not exist.

A: limx, y→2, 14-xyx2+3y2Plug in the value x, y=2, 1=4-2· 122+3· 12=27

Q: Which of the following statements are correct for C-4y lim (,1) (0,0) 2y +3 I) Approaching along the…

A: Let's move across the path y = x We get the value of -3/5 but that doesn't mean that limit exists.…

Q: 3, Use the definition of limit (E & 8) to show that lim (2.x+iy)-4i, where z=x+ iy. 21

A:

Q: Evaluate the limit lim z→2+i [(z−1)(3−2i)+1−3i] and use the definition of the limit (−δ) to…

A:

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

A: Consider the equation

Q: We have the function if (x, y) # (0,0), xy x²+y² f (x, y) if (x, y) = (0,0), (а) Find lim…

A: We have to find the following given conditions according to the given condition

Q: 6 Use the e - d definition of a limit to prove e lim 1+2 r2 + 2 1.

A:

Q: Prove that lim (r2- 1) = 3 using the e, 8 definition of the limit. %3D 141 Linh tL

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Q: Prove that the following limit does not exist: lim (z,9)(0,0) I-y

A: To Prove: lim(x,y)→(0,0)x+yx-y does not exists   Here, we will check the value of the limit along…

Q: evaluate the limit using continuity lim(x,y)→(−2,1)2x2/4x + y

A: To evaluate lim(x,y)→(-2,1)2x24x+y

Q: The limit lim (x,y) →(-2,-5) exists and is equal to -20. Select one: O True O False 25x²-4y² 5x-2y

A: The objective of the question is determine the limit existing.

Q: evaluate the limit using continuity. lim (x,y)→(1,1) x+y

A: limx,y→1,1ex2-e-y2x+y

Q: Use the e – 8 definition of the limit to prove that lim x2 – 2 = 2. %3D - x→2

A:

Q: Use the definition of the limit ( epsolon - delta ) to show that lim of 1/z as z approaches -i

A:

Q: Show that the limit does not exist. xy? lim (1,y)→(0,0) x³ + y6

A:

Q: 1) Show that the limit doesn't exists : 3x2 y lim (x,y)-(0,0) x2 + y2

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Q: Evaluate the limit using the Basic Limit Laws

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Q: lim -y3-3y²+4 y--2 -2y2-y+6

A:

Q: 2

A: According to the given information, it is required to prove the limit using epsilon delta…

Q: 8. Find the limit, if it exists, or show that it does not exist: 2x-y a) lim (xy)(1,2) 4x2-y2

A:

Q: 1. Find the limit if exists or show that the limit does not exist 3x²y² lim (,) (0,0) 24+24 y² (a)…

A: To find the limit if exists or to show that the limit does not exist.a limx, y→0, 0 3x2y2x4+y4b…

Q: Use the definition of the limit (e – 8) to show that 1 lim - = i. 2+-i z

A:

Q: The limit lim (x,y)→(4,-5) exists and equals to Select one: O True O False 25x²-16y² -5x-4y -40

A: Given the limit is  limx, y→4,-5 25x2-16y2-5x-4y.

Q: Show that the limit does not exist. xy? - 1 lim (Ky)→(1,1) y-1

A:

Q: Let 3a? -y f (x, y) = x² +y (1) Find the limit at (1, 1): lim(2,9)→(1,1) f(x, y). (2) Show that the…

A:

Q: 1 Use the definition of the limit (ɛ – 8) to show that 1 lim = i. z-i z

A: We will prove the given limit.

Q: Evaluate the limit below. 1 lim (x,y)→(0,0) 1-x2-y?

A: limx, y→0, 011-x2-y2

Q: Find the limit, if it exists, or explain that the limit doesn’t exist. (a) lim…

A:

Q: Use continuity to evaluate the limit. e lim (x,y)→(-4,–4) x + y

A:

Q: 2) Show that the limit exists : 3x3 ,3) lim (x.y.z)-(0,0,0) x2 +y² + z2

A:

Q: Find the limit lim,→+∞In(x² – 4) – In(x + 2)] Enter I for o, -I for –∞, and DNE if the limit does…

A: Given, limx->inf [ln(x²-4)-ln(x+2)]

Q: 1. Use the e - d definition of the limit to show the following: (a) lim(5x – 2) = 3 エ→1 (b) lim Vr =…

A: The objective is to use definition of limit to prove the above question.

Q: 8. Let f(x, y) be the function x3 + y3 xy? Show that the limit lim(9)→(0,0) f (x, y) does not exist…

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Q: Use the definition of "limit of f at c" to prove the limit of lim, 3(x² – 5x + 1) = -5

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

A: Given: limx→2x-2sin1x-2

Q: Verify using the definition of limit (with e and 8) that lim 1|+3) = 3, x1 xplain all the steps and…

A: To verify the below limit using the epsilon-delta definition. limx→112x−1+3=3

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!