Iterated LimitsDate: 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/ |
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