Find multivariable limits

Question

Find multivariable limits and explain

 

lim (x,y) -> (0,0) = (x^4-y^4)/(x^2+y^2)

 

lim (x,y) -> (0,0) = xy/sqrt(x^2+y^2)

Answer 1

For the first one,   x⁴ - y⁴  factors as (x² - y²)(X² + y²)  

the second factor cancels out leaving x² + y²    which is zero in the limit indicated

 

For the second one,   change top polar coordinates:    x = r cos(θ)  ,  y = sin(θ)  

the expression becomes   r² cos(θ) sin(θ) / r =  r cos(θ) sin(θ)    

  The limit as (x,y) approaches (0,0) forces r to zero.     The trig functions are bounded between -1 and 1 so

  the limit is zero