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: In Exercises 27-42, evaluate the limit or determine that it does not exist.

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Q: Let f(x) = x²sin () for x + 0 and f (0) = 0. b) Use definition 6.1.1 to show that fis differentiable…

A: Solving by definition of 6.1.1 we have

Q: lim f(x, y) = L and f is defined at (a, b), then f(a, b) = L. If %3D (z,3)→(a,b) O True O False

A: Let us define a function:f(x,y)=3x+4y , (x,y)≠05, (x,y)=(0,0)This shows that…

Q: 1) Let f: R³ R be defined by, f(x, y, z)= 3x - 5y + 2z. a) Prove by using an e-8 definition the…

A: The given function  f: ℝ3 →ℝ be defined by,                                f(x,y,z) = 3x-5y+2z.

Q: Let f : [0, ∞) → R be differentiable everywhere. Assume that limx→∞  (f(x) + f’(x) )= 0. Show that…

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Q: 6. Determine the limits obtained by approaching (0,0) along both the x and y What does this say…

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Q: Evaluate lim (z,y) → (0,0) 1 - cos(xy) x²y²( −3+ y)

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Q: Evaluate the limit for r(t) = (-8, sin(t), 8). r(t + h) - r(t) h lim h→0 (Use symbolic notation and…

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Q: P (x) = x² + 12x³ + 12x² + 12x + 11 = 0

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