Let f ( x , y ) = ( x 2 </msup> + y 2 </msup> + 2

Question

Let

f (x, y) = (x^{2} + y^{2} + 2)^{x y}

defined on

R^{2}

and let

g (x, y)

be a degree 2 polynomial in two variables.
Suppose

lim_{(x, y) \to (0, 0)} \frac{g (x, y) - f (x, y)}{x^{2} + y^{2}} = 0

. Then, determine

g (x, y)

Answer 1

Recall that for

e^{t} = 1 + t + O (t^{2})

and

\ln (1 + t) = t + o (t)

. Therefore,for

(x, y) \to (0, 0)

\begin{aligned} (x^{2} + y^{2} + 2)^{x y} & = \exp (x y \ln (x^{2} + y^{2} + 2)) \\ = 1 + x y \ln (x^{2} + y^{2} + 2) + O (x^{2} y^{2}) \\ = 1 + x y (\ln (2) + \ln (1 + \frac{x^{2} + y^{2}}{2})) + o (x^{2} + y^{2}) \\ = 1 + x y (\ln (2) + \frac{x^{2} + y^{2}}{2} + o (x^{2} + y^{2})) + o (x^{2} + y^{2}) \\ = 1 + \ln (2) x y + o (x^{2} + y^{2}) . \end{aligned}

Hence

g (x, y) = 1 + \ln (2) x y

. In particular

a_{3} = \ln (2)

Answers (1)