10. Let ƒ(2) = { (24/3/5/3 + ix³/34/3)/(x² + y²), if z ‡ 0; 0 if z = 0 Show that the Cauchy-Riemann equations hold at z = 0 but f is not differentiable at z = 0. Hint: consider the limit with Az = (1 + i)h, h→0.

Q: Q Find the derivativeof y)with respect tox; in the foltowina function s and u=3X 2 Find dy for the…

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Q: Evaluate the following statement: the function z+y+z f(z,y, z) := { z+v+zl' * +y+z+ 0, 0, z + y + z…

A: The function f(x,y,z)=x+y+zx+y+z, x+y+z≠00,                   x+y+z=0, we have to check the…

Q: D) How an un 04 A ) Use sandwich theorm to find the limit of the function g(x) as x approach zero.…

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Q: 2. The limit as x→ 0 of sin x X

A: Given limx→0sinxx .

Q: 2 2. Express the limit lim E(-1+4): as a definite integral. DO NOT Evaluate it. i=1

A: The given limit is: limn→∞∑i=1n-1+2in2n To express the given limit as a definite integral.

Q: 16. If f is differentiable at xo, prove that lim h→0 f(xo + αh) - f(xo - ßh) h = (x + ß)ƒ'(xo).

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Q: 3. Show that lim(r.y)→(0,0) xy =0. (Hint: Use the fact the y| = Vy² and therefore y|< Vx² + y²)

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Q: gt Voy +9/1 + Y Y (1 - e-t).

A: We have to find

Q: 24. If the second derivative f" exists at a value xo, show that f(xo + h) - 2f(xo) + f(xo − h) h²…

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Q: 12.Find f(Vx), given the transcendental function f(x)=in x

A: The given transcendental function is: fx=lnx

Q: 3. For m₁ and m₂ numbers, with m₁ { Prove that the function f:R → R is continuous but not…

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Q: Let z = 1+ iy. Determine whether f(2) is entire function or not? (Hint: Use Cauchy- Riemann…

A: Let  f(z) = u(x, y) + iv(x, y) Then the Cauchy-Riemann equations are ∂u∂x=∂v∂y and ∂u∂y=-∂v∂x

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A: To prove

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Q: 15. Suppose that the functionf:R Ris differentiable at xg. Prove that xf (xo)- xof (x) lim = f(xo) -…

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Q: 2. Suppose that f is defined on (0, b) and that f is differentiable at c € (0, b). (Do not assume…

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