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Limits and Continuity. Definition: Limit of a Function of Two Variables. Let f be a function of two variables defined, except possibly at (x o , y o ), on an open disk centered at (x o , y o ), and let L be a real number. Then:. If for each ε > 0 there corresponds a δ > 0 such that:. - PowerPoint PPT Presentation
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Limits and Continuity
Definition: Limit of a Function of Two Variables
Let f be a function of two variables defined, except possibly at (xo, yo), on an open disk centered at (xo, yo), and let L be a real number. Then:
Lyxf
oo yxyx
),(lim
,),(
If for each ε > 0 there corresponds a δ > 0 such that:
220 whenever ),( oo yyxxLyxf
1. Use the definition of the limit of a function of two variables to verify the limit
(Similar to p.904 #1-4)
3lim)0,3(),(
xyx
2. Find the indicated limit by using the given limits(Similar to p.904 #5-8)
),(),(3lim:
2),(lim
,8),(lim:
),(),(
),(),(
),(),(
yxgyxfFind
yxg
yxfGiven
bayx
bayx
bayx
3. Find the limit and discuss the continuity of the function
(Similar to p.904 #9-22)
xyxyx
2lim 2
)1,4(),(
4. Find the limit and discuss the continuity of the function
(Similar to p.904 #9-22)
1
lim2
)2,5(),( y
xyx
5. Find the limit and discuss the continuity of the function
(Similar to p.904 #9-22)
)sin(lim 2
,4
1),(
xyxyx
6. Find the limit and discuss the continuity of the function
(Similar to p.904 #9-22)
)arcsin(lim
2,2),(yx
yx
7. Find the limit (if it exists). If the limit does not exist, explain why
(Similar to p.905 #23-36)
yx
yxyx 3
9lim
22
0,0),(
8. Find the limit (if it exists). If the limit does not exist, explain why
(Similar to p.905 #23-36)
yx
yx 0,0),(lim
9. Find the limit (if it exists). If the limit does not exist, explain why (Hint: think along y = 0)
(Similar to p.905 #23-36)
yx
yyx
1lim
0,0),(
10. Discuss the continuity of the functions f and g. Explain any differences
(Similar to p.906 #43-46)
)0,0(),(,2
)0,0(),(,3
9),(
)0,0(),(,0
)0,0(),(,3
9),(
22
44
22
44
yx
yxyx
yxyxg
yx
yxyx
yxyxf
11. Use polar coordinates to find the limit [Hint: Let x = r cos(θ) and y = r sin(θ), and note that
(x, y) -> (0, 0) implies r -> 0](Similar to p.906 #53-58)
22
2
0,0),(
3lim
yx
yxyx
12. Use polar coordinates and L’Hopital’s Rule to find the limit
(Similar to p.906 #59-62)
22
22
0,0),(
)sin(6lim
yx
yxyx
13. Discuss the continuity of the function(Similar to p.906 #63-68)
5),,(
2
3
yx
zzyxf
14. Find each limit(Similar to p.906 #73-78)
y
yxfyyxfb
x
yxfyxxfa
yxyxf
y
x
),(),(lim)
),(),(lim)
3),(
0
0
22
