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### Iterated Limits

```
Date: 5/25/96 at 11:30:10
From: Joel Mendez
Subject: Iterated Limits

Hi Dr. Math,

don't know how to do the following problems:

{ x*sin(1/y)    if  y<>0
f(x,y)={
{     0         if  y=0

a) Demonstrate that
Lim (Lim f(x,y)) <> Lim (Lim f(x,y)
x->0 y->0           y->0 x->0

b) Demonstrate that
Lim f(x,y) = 0
(x,y)->(0,0)

c) Why does it not contradict the theorem of the iterated limits?

Early thanks...
Joel Mendez
```

```
Date: 5/29/96 at 18:59:5
From: Doctor Pete
Subject: Re: Iterated Limits

The key to this whole problem is the behavior of sin(1/y) for small y.
Note that the sine is always bounded between -1 and 1, so no matter
what (nonzero) value you choose for y, sin(1/y) will always be between
-1 and 1.

Part a) the right side of the expression is therefore obviously 0.
But the limit on the left hand side doesn't exist. Ask yourself what
lim f(x,y), y->0 is, and remember that an iterated limit is evaluated
step by step.

Part b) is a bit more difficult, but the tricky thing to remember here
is that this limit is taken with x 'and' y going to 0 'at the same
time'. Since sin(1/y) always stays between -1 and 1, what can you say

Part c) is about the existence of iterated limits: If lim f(x,y) = L
for {x->a, y->b}, and the individual limits lim f(x,y) {x->a},
lim f(x,y) {y->b} also exist, then:

lim(lim f(x,y) {x->a}) {y->b}) = lim(lim f(x,y) {y->b}) {x->a}) = L.

So part b) tells us L = 0, but part a) tells us the iterated limits
are not equal.  What's wrong?  (Hint: look at the individual limits,
in particular, what is lim f(x,y), y->0?)

-Doctor Pete,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus

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