Math 234 - Fall 2024
Worksheet 14.2
[Review]
Let f (x, y) be a function of two variables.
1. Limits: We write
lim
f (x, y) = L
(x,y)−→(a,b)
if f approaches L as (x, y) −→ (a, b) along any path.
2. Continuity: f is called continuous at (a, b) if
lim
f (x, y) = f (a, b).
(x,y)−→(a,b)
3. Polynomials (e.g. x2 y + xy + 3x5 y 6 ) are continuous everywhere.
2
3
) are continuous at every point in their domain.
4. Rational functions (e.g. y +xy+x
x4
5. To show that a limit does not exist, we need to find two different paths that give two different limits.
[Problems]
1. Most of the following limits won’t exist, but some of them will. If the limit doesn’t exist, show how
you know that it doesn’t. If it does exist, find the value of the limit.
(a)
x
.
(x,y)→(0,0) x + y
(b)
x2 + y 2
.
(x,y)→(0,0) x + y 2
(c)
x sin(y)
.
y2
(x,y)→(0,0)
(d)
lim
lim
lim
lim
xy 2
(x,y)→(0,0) xy 2 + 3x2 y 4
.
1
Math 234 - Fall 2024
(e)
(f)
x2
.
(x,y)→(0,0) y + y 3
lim
lim
4
x sin
(x,y)→(0,0)
(g)
lim
x3 y cos
(x,y)→(7,0)
(h)
lim
(x,y)→(0,2)
2
1
.
x2 + y
1
.
xy 2
3
x (y − 1) sin
1
.
x2
2