Bard Math 212 - Multivariable Limits

557 days ago by ethan

A Limit That Does Not Exist

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

x, y = var('x,y') plot3d((x^2 - y^2)/(x^2 + y^2), (x,-4,4), (y,-4,4), aspect_ratio=1)

Another Limit That Does Not Exist

\displaystyle \lim_{(x, y) \to (0, 0)} \frac {xy}{x^2 + y^2}

x, y = var('x,y') plot3d((x*y)/(x^2 + y^2), (x,-4,4), (y,-4,4), aspect_ratio=1)

A Limit That Does Exist

\displaystyle \lim_{(x, y) \to (0, 0)} \frac {3x^2y}{x^2 + y^2} = 0

x, y = var('x,y') plot3d((3*x^2*y)/(x^2 + y^2), (x,-4,4), (y,-4,4), aspect_ratio=1)