Limits and Continuity
Reading assignment: 2.1, 2.2
Recommended problems: 2.1: 1, 3, 5, 7, 14, 19, 21, 27, 28, 31, 32, 33, 34, 36 – 42
2.2: 9, 11, 13, 15, 17, 22, 24 – 27, 28, 29, 31, 33, 41, 46. 48
For each function below, either provide a value for f(0, 0) that will make the function continuous
or explain why no such value exists. Justify your results.
1) f ( x, y ) 
2) f ( x, y) 
x
x  y2
2
x3  x 2  xy 2  y 2
x2  y 2
3) f ( x, y ) 
xy
x y
4) Carefully sketch some of the level curves of g ( x, y)  y 2  x 2 . Attempt to sketch the surface.
Keeping x and y in a range of [ 3,3] should be fine.
y
3
x
-3
-3
3
5) Find an equation for the tangent plane to the surface z  x 2  y 2 at an arbitrary point (a, b, c)
on the surface. Sketch.
6)
Evaluate the following limit, or show that the limit does not exist.
a.
b.
lim
( x , y ) (0,0)
 sin( xy ) 
 x2  y 2 


Lim [( x 2  y 2 ) Ln( x2  y2)]
(x , y ) (0,0)