4. Let f(x, y) = y ln(x² + y²) whenever (x, y) = (0,0) and f(0,0) = 0. Show that f, is defined at all points (x, y), and that fe is not continuous at (0,0).

Q: Let f:R2 →R be defined by f (x, y) = yiif (x,y) # (0,0) 0, if (x,y) = (0,0) continuous at (0,0).…

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A: Given function is z=gx,y=x5−y5x4+y4 for x,y≠0,00            for x,y=0,0. We have to check whether…

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Q: Show that A Function with a Single Point of Discontinuity 2xy x² +1,²² 0, is continuous at every…

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Q: Let f(x, y) = and let g(x, y) O O rºy6 26+y6 O 0 0 6x³y³ 6x6 + 6y6 Which of the above functions are…

A: fx,y=x6y6x6+y6if x,y≠0,00if x,y=0,0 gx,y=6x3y36x6+6y6if x,y≠0,00if x,y=0,0

Q: z = f(a + y, 2ry) f we take u = x' + y' and w = 2xy we obtain z, = 2xf, + 2yf, and z, = 2yf, + 2xf.,…

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Q: Let z=x²y, where x(s,t) and y(s,t) are differentiable functions with values Әx Əs x(2,0) = 4, y(2,0)…

A: Given : z = x^2.y

Q: Assume that f(x, y) is a function given by if (r, y) # (0,0), f(x, y) 0 if (r, y) = (0, 0). Page 1…

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Q: { 2xy? 9x2 + y4 (x, y) # (0,0) (x, y) = (0,0) f (x, y) ·

A: Topic = Limits and  Continuity

Q: Assume that f(r, y) is a function given by if (r, y) # (0,0), f(r, y) = 0 if (r, y) = (0, 0).

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Q: What value of c makes the following function continuous at (0,0)? f(z, y) = C, x+ y³ +2, (x,y) +…

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Q: Let f(x, y) = x/y. Then the directional derivative of f in the direction of √2 point (1, 1) is ○ (1,…

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Q: a sin (), it (, u) + (0, 0) x²+y? 2. Determine if a function f : R² → R defined by f (x, y) = 0, if…

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Q: Calculate of of and for the function f(x, y) defined as : ax' dy sin (xy) 1 + y f(x, y) =

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Q: d) Let f be a continuous function. Then 1 [ S f(x,y) dady = f*f* f(x,y) dyda. 0

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Q: Show that the function g(x, y) = is not continuous at (0,0). xy 2x² + y²¹ 0, (x, y) = (0,0) (x, y) =…

A: g(x,y)=(xy/(2x2+y2), (x, y) not equal to (0,0)               0                 ,(x, y) = (0,0)

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Q: 15. Let f(x, y) = 22 if (x, y) ‡ (0,0), and f(0,0) = 0. Is ƒ continuous at (0,0)? x+y

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