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Iterated Limits


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

Hi Dr. Math, 

I study in the Unimet in Caracas, Venezuela.  Please help me because I 
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 
about x*sin(1/y) as x->0?

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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