Dete‎rmine if f (x, y) is‎ co‎ntinuous at (0,‎ 0)

Question

Dete‎rmine if f (x, y) =

={ (xsi‎n³y)/(x‎²+y⁶), (x, ‎y) ≠ (0, ‎0)

={ 1, (x, y) = (0, 0)

is‎ co‎ntinuous at (0,‎ 0)

Answer 1

It is not continuous because lim(f(x,y)) as (x,y)->(0,0) is not equal 1.

(Actually the limit does not even exist.)

To see that the limit cannot be equal 1, considering approaching (0,0)

along the curve

y = x^1/3

On that curve, f becomes a function of the single variable x

f(x) = (xsin³x^1/3)/(x²+(x^1/3)⁶) =

(xsin³x^1/3)/(2x²) =

(sin³x^1/3)/(2x)

As x->0, lim(f(x)) = ∞