u = tan–¹ (x²+y²/x–y)

1. Verify Eular's theorem for the function

u = tan–¹ (x²+y²/x–y)

Sol:

u = tan–¹ (x²+y²/x–y)

tan u = (x²+y²/x–y) = f

n = 3 -1 = 2

By Eular's theorem for homogeneous function f in x and y

x ∂f/∂x + y ∂f/∂y = n.f

x ∂/∂x (tan u) + y ∂/∂y(tan u) = 2x tan u

x sec²u ∂u/∂x + y sec²u ∂u/∂y = 2tan u

x ∂u/∂x + y ∂u/∂y = 2tan u/ sec²u

x ∂u/∂x + y ∂u/∂y = 2sin u/cos u ÷ 1/cos²u

x ∂u/∂x + y ∂u/∂y = 2sin u x cos²u/cos u

x ∂u/∂x + y ∂u/∂y = sin 2u

Proved 

Note:

{ 2sinacosa = sin2a} 

2. lim x tends a [ x^n - a^n /x-a]

Sol:

Putting x = a = (0/0)

By L' hospital rule

lim x tends a ( nx^n-1 - 0 / 1- 0)

= na^ n-1

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