March 20th, 2019, 03:11 AM  #1 
Newbie Joined: Mar 2019 From:  Posts: 3 Thanks: 0  Multivariable calculus  Limits
Show that the function f(x,y)=y/(xy) for x→0, y→0, can take any limit. Construct the sequences { f(xn, yn } with (xn,yn)→(0, 0) in such way that the lim n→∞ f(xn,yn) is 3,2,1,0,−2. Hint: yn=kxn. I am not sure whether I am right, but I did the following: f(x,y) = kxn/(xn−kxn) = kxn/(xn(1−k)) = k/(1−k) k/(1−k) = A After a few steps I have obtained: k = A/(1+A), A ≠ −1 So, the function can take any limit except 1. Now we have: A {3,2,1,0,−2} If A=3: k=3/4, yn=3/4xn So lim n→∞ (3/4xn)/(1/4xn)=3. For A = 2, 1, 0 and −2, I did the same. I am not sure whether this is the end of the exercise. I don't understand what exactly in this context is meant by the sequence. I am also not sure whether I should write lim (xn, yn)→(0,0) (3/4xn)/(1/4xn)=3 instead of lim n→∞ (3/4xn)/(1/4xn)=3 or just xn→0, because I have just xn. Any help is appreciated. Thanks in advance. Last edited by skipjack; April 8th, 2019 at 03:47 PM. 
March 21st, 2019, 11:20 AM  #2 
Senior Member Joined: Dec 2015 From: somewhere Posts: 605 Thanks: 88 
If you change the order of limits you get $\displaystyle 1=\lim_{x\rightarrow 0} \frac{0}{x}$ which is not true .
Last edited by idontknow; March 21st, 2019 at 11:25 AM. 
March 21st, 2019, 11:52 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra  You were asked to show that it can take any limit. How are you going to get 1? Hint: approach along the $y$ axis. Having shown, for example, that with $y_n=\frac34 x_n$ you get $\displaystyle \lim_{n \to \infty} f(x_n,y_n) = 3$, you should interpret what $y_n=\frac34x_n$ means in terms of the approach to $(0,0)$. In this case you are approaching the origin along the straight line $y = \frac34x$. This is the answer that I consider most appropriate, although the talk of sequences may mean that your teacher has something else in mind. 
March 21st, 2019, 11:54 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra  

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calculus, limits, multivariable, sequence 
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