What is the limit as (x,y) approaches (0,0) of (x^2+y^2)ln(x^2+y^2) ?

The limit as (x,y) approaches (0,0) of (x^2+y^2)ln(x^2+y^2) is 0^2ln(0^2+0^2)+0^2ln(0^2+0^2)

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limit as (x,y) approaches (0,0) of (x^2+y^2)ln(x^2+y^2)

(x, y) \to (0, 0) lim ((x^{2} + y^{2}) ln (x^{2} + y^{2}))

0^{2} ln (0^{2} + 0^{2}) + 0^{2} ln (0^{2} + 0^{2})

Solution steps

(x, y) \to (0, 0) lim ((x^{2} + y^{2}) ln (x^{2} + y^{2}))

Factor out

x^{2}

from

x^{2} + y^{2} : x^{2} (1 + \frac{y ^{2}}{x ^{2}})

= (x, y) \to (0, 0) lim (x^{2} (1 + \frac{y ^{2}}{x ^{2}}) ln (x^{2} + y^{2}))

Simplify

x^{2} (1 + \frac{y ^{2}}{x ^{2}}) ln (x^{2} + y^{2}) : x^{2} ln (x^{2} + y^{2}) + y^{2} ln (x^{2} + y^{2})

= (x, y) \to (0, 0) lim (x^{2} ln (x^{2} + y^{2}) + y^{2} ln (x^{2} + y^{2}))

Plug in the value

(x, y) = (0, 0)

= 0^{2} ln (0^{2} + 0^{2}) + 0^{2} ln (0^{2} + 0^{2})

What is the limit as (x,y) approaches (0,0) of (x^2+y^2)ln(x^2+y^2) ?
The limit as (x,y) approaches (0,0) of (x^2+y^2)ln(x^2+y^2) is 0^2ln(0^2+0^2)+0^2ln(0^2+0^2)