Math 237
Final Review Problems
1.
Domain/Level Curves/Crosssections
2.
Limits/Continuity/Diﬀerentiability
a) Write the precise deﬁnition of the limit.
Solution: We write
lim
(
x,y
)
→
(
a,b
)
f
(
x,y
) =
L
if for every
± >
0 there exists a
δ >
0 such that

f
(
x,y
)

L

< ±
whenever 0
<
k
(
x,y
)

(
a,b
)
k
< δ
.
b) How do you try to evaluate a limit?
Solution: First, see if we can observe the function is continuous. Otherewise, try to
prove it doesn’t exist by approaching along diﬀerent lines and curves. If we get two
limits with diﬀerent values then the limit doesn’t exist. If all lines and curves give the
same value
L
then we try to prove the limit exists using the squeeze theorem. If we
can not make the squeeze theorem work, then we must ﬁnd two curves giving diﬀerent
values to prove the limit doesn’t exist.
c) What is the deﬁnition of continuous.
Solution:
f
(
x,y
) is continuous at (
a,b
) if
lim
(
x,y
)
→
(
a,b
)
f
(
x,y
) =
f
(
a,b
).
d) What is the deﬁnition of diﬀerentiability.
Solution:
f
(
x,y
) is diﬀerentiable at (
a,b
) if
1.
f
x
(
a,b
) and
f
y
(
a,b
) both exist. 2.
lim
(
x,y
)
→
(
a,b
)

f
(
x,y
)

L
(
a,b
)
(
x,y
)

k
(
x,y
)

(
a,b
)
k
= 0.
e) What are two important theorems about diﬀerentiablity.
Solution: If
f
x
and
f
y
are both continuous at (
a,b
) then
f
is diﬀerentiable at (
a,b
). If
f
is diﬀerentiable at (
a,b
) then
f
is continuous at (
a,b
).
3.