Limit when x^2 + y^2 -> inf, am I solving it correctly?

  • Thread starter Addez123
  • Start date
  • #1
Addez123
184
15
Homework Statement:
Decide if the limit exists.
Relevant Equations:
$$\lim_{x^2+y^2 \rightarrow +\infty} {\frac {xy} {x^2+y^2}}$$
I'm not sure if the way I solve these limits is correct, so let me know if I'm doing something wrong.
$$\lim_{x^2+y^2 \rightarrow +\infty} {\frac {xy} {x^2+y^2}}$$
$$r = x^2+y^2$$
$$\lim_{r \rightarrow +\infty} {\frac {r\cdot cos(v) \cdot r \cdot sin(v)} r}$$
$$\lim_{r \rightarrow +\infty} {r\cdot cos(v)sin(v)} \rightarrow \infty$$


Also how can I be sure if $$cos(v)\cdot sin(v)$$ will make this go to inf, -inf or even be undefined?
I havn't even defined v..
 

Answers and Replies

  • #2
BvU
Science Advisor
Homework Helper
15,269
4,245
I havn't even defined v..

Not only that, you define ##r = x^2+y^2## which means ##x \ne r\cos\theta##
Try again, now with ##x =\sqrt r\cos\theta##

And think about what the role and meaning of ##\theta## could be ...
 
  • #3
Addez123
184
15
Ah, my bad. I should've defined it as: $$r^2 = x^2 + y^2$$
$$\lim_{x^2+y^2 \rightarrow +\infty} {\frac {xy} {x^2+y^2}}$$

$$\lim_{r^2 \rightarrow +\infty} {\frac {r\cdot cos(v) \cdot r \cdot sin(v)} {r^2}}$$
$$\lim_{r^2 \rightarrow +\infty} {cos(v)sin(v)}$$

Well, this just made it trickier. I have no idea what the angle is.
v = atan(y/x) doesn't really help..

Should I have solved this in a completely different way?
 
  • #4
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2022 Award
23,784
15,394
Ah, my bad. I should've defined it as: $$r^2 = x^2 + y^2$$
$$\lim_{x^2+y^2 \rightarrow +\infty} {\frac {xy} {x^2+y^2}}$$

$$\lim_{r^2 \rightarrow +\infty} {\frac {r\cdot cos(v) \cdot r \cdot sin(v)} {r^2}}$$
$$\lim_{r^2 \rightarrow +\infty} {cos(v)sin(v)}$$

Well, this just made it trickier. I have no idea what the angle is.
v = atan(y/x) doesn't really help..

Should I have solved this in a completely different way?

Whenever you are stuck like this you should ask: what is the definition of the thing I'm trying to prove? In this case, what does this mean (in precise mathematical terms):
$$\lim_{x^2+y^2 \rightarrow +\infty} {\frac {xy} {x^2+y^2}}$$
 
  • #5
Delta2
Homework Helper
Insights Author
Gold Member
5,695
2,473
I think your work (after the correction regarding r) is good. You find that the limit is ##\cos v\cdot\sin v## which implies that it depends on ##v## which implies that the limit does not exist.
 
  • #6
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2022 Award
23,784
15,394
I think your work (after the correction regarding r) is good. You find that the limit is ##\cos v\cdot\sin v## which implies that it depends on ##v## which implies that the limit does not exist.

Alternatively, of course, you could choose a sequence of points with ##x = 0## and ## y \rightarrow \infty##, where the limit for that sequence is ##0##. Then, a sequence of points with ##x =y##, where the limit is ##1/2##.

From these examples alone you can see the limit does not exist.
 
  • Like
Likes Addez123 and Delta2
  • #7
BvU
Science Advisor
Homework Helper
15,269
4,245
Well, this just made it trickier. I have no idea what the angle is.
v = atan(y/x) doesn't really help..
But it should ! ##\ ## I had silently hoped that this would have reminded you of another non-existing limit, $$\lim_{(x,y) \downarrow (0,0)}\; {y\over x}$$ where ##{y\over x}## similarly has a value that depends on the angle at which you approach the origin...

Now that you have been given the answer, is it clear ?
 
  • #8
Addez123
184
15
I think what @PeroK did is what they expect us to do. It's familiar to previous exercises.
@BvU I don't see how I would've solved for x and y to get v without going in circles like:
$$v = atan( \frac {r \cdot sin(v)}{r\cdot cos(v)})$$
 
  • Like
Likes Delta2 and BvU
  • #9
PeroK
Science Advisor
Homework Helper
Insights Author
Gold Member
2022 Award
23,784
15,394
I think what @PeroK did is what they expect us to do. It's familiar to previous exercises.
@BvU I don't see how I would've solved for x and y to get v without going in circles like:
$$v = atan( \frac {r \cdot sin(v)}{r\cdot cos(v)})$$
You would do the equivalent of what I did but in polar coordinates. ##v = \pi/2## equates to ##x = 0##; and ##v = \pi/4## equates to ## x = y##.
 

Suggested for: Limit when x^2 + y^2 -> inf, am I solving it correctly?

Replies
3
Views
694
Replies
26
Views
1K
Replies
21
Views
973
Replies
24
Views
618
  • Last Post
Replies
6
Views
472
  • Last Post
Replies
3
Views
412
  • Last Post
Replies
9
Views
306
Replies
12
Views
464
Replies
7
Views
476
  • Last Post
2
Replies
54
Views
3K
Top