QUESTION 4 ху? Let f(x) = { Jx²+y? if (x.y) # (0,0) if (x,y)=(0,0) lim f(x.y). (x.y) → (0,0) a) Evaluate b) Is the function f continuous at (0,0)? Justify.

Q: Show that there does not exist a function f (x, y) such that ∇f =〈y^2, x〉. Hint: Use Clairaut’s…

A: To show that there does not exist a function f(x,y)

Q: Theorem 7. Let X be a continuous RV, g a function from R to R. Let Y = g(X). Then E[Y] = E[g(X)] = /…

A: The expected value of random variable is the weighted average of all possible values of the…

Q: lim f(x, y) does not exist, where (1,4)→(0,0) Show that f(r,y) (x² + y1)³

A:

Q: tan 3h 3. Estimate the value of lim,0- by creating a table of function values for h h = 0.01,0.001,…

A: We need to substitute these values of h in function f(h) to evaluate limit

Q: I. Show that the function f defined by 1, (x, y) = (1, – 1) f(x, y): x2 + y (x, y) # (1, – 1) x +y…

A:

Q: f:(-m,m)-R, f(x)=xt cosx, (MER") function, at the point of a t (-M,m) COSX Ts this function uniform…

A:

Q: (a) Is the function f(x, y) = xy(x²-x²) when (x, y) = (0,0) and f(0, 0) = 4 continuous at origin.…

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Q: 2) Show that sinax _a lim x-0 tanbx b %3D

A: Show that  limx→0sinaxtanbx

Q: 1 Let f (x, y) =x²y*sin (- x²+y* when (x, y) ± (0,0) and f (0,0) =0. determine if f is continuous in…

A: We have to show given function is continuous at origin

Q: The definition of the first derivative of a function f(x) is: Select one: f(x+Ax)-f(x) a. f'(x) =…

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Q: if (x, y) # (0,0) 5) Is f(x, y) = 0 if (x, y) = (0,0) continuous at (0,0)? Explain.

A: In calculus, a continuous function is a real-valued function whose graph does not have any breaks or…

Q: Explain why this statement is false The function g(x) = x^2 on the interval [-1, 1] has the…

A:

Q: 5. Define f : (0, 1) → R by f(x) = x+1 Can one define f(0) to make f continuous Vx at 0? Explain.

A: Note: As per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: (a) Find the equation of the tangent line to the curve 4x y-3x2 +5y = 10, at the point (1,-1).…

A: a)

Q: 3. What are is the absolute maximum of f (x) = sin(x) cos(x) on the interval [0, 1]?

A: follow next step

Q: Use Theorems 1-5 to show that the function zs continuous. f(x) = x + sinx

A: Consider the given function: fx=x+sinx As we know, sum of finite continuous function is continuous.…

Q: 5. If lim f(x, y) = 5 (x,y)→(3,–7)" %3D for every curve C in R? containing the point (3, -7), what…

A: If limx,y→3,−7fx,y=5  is true, then it represents that limit exists atpoint (3,-7) for function…

Q: 4. Prove that a function which is harmonic everywhere on a plane and is bounded either above or…

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Q: Let f: [2,3] → {1,3} be a continuous function such that the equation f(x) - x = 0 has a solution in…

A:

Q: The limit represents the derivative of some function f at some number a. State such an f and a. lim…

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Q: hey you didnt state the assumptions of the model and the limintations as well

A: To state: Assumptions and limitations of an exponential model.

Q: ((2x-y2) Let f(x, y) = x² + y² (x, y) = (0,0) 2, (x, y) = (0,0) Are fx and fy continuous at (0,0)?…

A: Given that f(x, y)=(2x-y2)2x2+y4,(x, y)≠(0, 0)2,(x, y)=(0, 0)

Q: Example Prove that sin(xy?) x2 + y? ' 0, if (x, y) # (0, 0), f(x, y) = if (x, y) = (0,0) is…

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Q: 5. Use the limit definition of the derivative to find the derivative of f(x) = 3x-2. f(x+h) - f(x)…

A: f'(x)=limh→0 f(x+h)-f(x)h

Q: กำหนดให้ f(x, y) = xy cos.x x² +3y²

A:

Q: II. Evaluate the limits. tan 0 2. lim 0 -0 sin 20 (y-#)* 2. 4. lim t 1-sin y y - 2 2.

A:

Q: 2. Determine the points where f is continuous where x + Y if (x, y) # (0,0) f(x, y) = x2 + y2 if (x,…

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Q: 3) Examine the limit of 1 (x² +y²).sin- x*+ (x, y) (0,0) f(x, y) = (0, (x, y)=(0,0)

A: Solution:Givenf(x,y)=x2+y2 sin 1x2+y2To Eamine the limit of a functionFormula:limx→0 sin xx=1

Q: 7. Let f(2) = f(x, y) = + i , = 0. when z +0 and let f(0) %3D 5xº+y² (a) Show that lim f(z) = f(0) =…

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Q: |h(x, y)| <lul show that lim h(r, y) = 0

A: Simple limit problem for multiple variable....

Q: I. Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = x2 + y (x, y) # (1, –1) x +y is…

A:

Q: 2) タx+y2 x²+y2 (x,y)#(0,0) (x,y) = (0, 0) . Let f(x y= ノ ノ Is f Continuous at 10,0) ? why ?

A:

Q: 3 (e) lim (1 x2

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Q: Determine if a function f : R2 → R defined by f (x; y) = { x sin(1/x2+y2) , if (x, y) ≠ (0,0) and 0…

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Q: 6. Prove if the following function is differentiable y2 = (arctanx + 1)°*

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Q: Question 5- Evaluate the limit. Vx-1 lim Question 6 - Find the equation of the tangent line to y = -…

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Q: Find lim (e-2x cos x) (Hint: Squeeze Theor

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Q: x*y° lim (x,y)→(0,0) f(x, y) does not exist, where f(x, y) = Show that (x2 + y4)3

A: To show that the below limit does not exist. lim x,  y→0,  0x4y4x2+y43

Q: Consider a function f continuous and differentiable on R with f(2. 5) = -3. 1 and f'(2. 5)=-0. 2.…

A: We have to find

Q: Let f [0, 1] → [0, 1] be continuous. Show that there exists an element x = [0, 1] such that f(x) =…

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Q: Evaluate using the Squee ze Theorem.

A:

Q: I. Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y…

A:

Q: You may assume that the sine and cosine functions are continuous and may freely use techniques from…

A: (a).   We have to find the limit of:   limx→0 sin(2x)-2xx3

Q: Exercise 1. Express the limit as a definite integral on the given interval a) lim T? cosxi Li=1' Ax,…

A:

Q: (b) Show that the function defined by √₂²+y f(x, y) = 0 if (x, y) = (0,0) is not differentiable at…

A: On solving this we will get

Q: Check the validity of Cauchy’s mean value theorem for the functions f(x)=x3 and g(x)=arctanx on the…

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Q: EXAMPLE 2 Find af/ay as a function if f(x, y) = y sin xy.

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Q: It f:[0,1]→R is a function that achieves a global minimum at 1, and we assume that limx+1- f(x) -…

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Q: If f is positive-valued continuous decreasing function and lim - S(x) = 0 then J F(x)dx diverges…

A:

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.