Using polar coordinates, show that lim (x,y)->(0,0) [sin(x^2+y^2)]/[x^2+y^2] = 1

In summary, using polar coordinates, we can show that the limit of [sin(x^2+y^2)]/[x^2+y^2] as (x,y) approaches (0,0) is equal to 1. Attempting to use L'Hospital's rule resulted in a messy and unhelpful expression. However, by letting ##\theta = r^2##, we can see that the limit is equivalent to one that has already been studied.
  • #1
bubbers
6
0

Homework Statement



Using polar coordinates, show that lim (x,y)->(0,0) [sin(x^2+y^2)]/[x^2+y^2] = 1

Homework Equations



r^2=x^2+y^2

The Attempt at a Solution



I was able to get the limit into polar coordinates:

lim r->0^+ [sin(r^2)]/r^2

but I'm not sure how to take this limit. I tried L'Hospital's, but it was just messy and useless seeming. So, any hints?
 
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  • #2
What went wrong when you tried to use L'Hospital's rule? Where did you get stuck?
 
  • #3
I got:

lim r->0^+ [sin(r^2)(2r)-r^2cos(r^2)(2r)]/r^4

which simplifies to:

lim r->0^+ [2sin(r^2)-2r^2cos(r^2)]/r^3

but that's still 0/0 form, so it got me nowhere useful as far as I can see.
 
  • #4
bubbers said:

Homework Statement



Using polar coordinates, show that lim (x,y)->(0,0) [sin(x^2+y^2)]/[x^2+y^2] = 1

Homework Equations



r^2=x^2+y^2

The Attempt at a Solution



I was able to get the limit into polar coordinates:

lim r->0^+ [sin(r^2)]/r^2

but I'm not sure how to take this limit. I tried L'Hospital's, but it was just messy and useless seeming. So, any hints?
Applying L'Hôpital's rule gives you:
[itex]\displaystyle \lim_{r\,\to\,0^{+}}\frac{\sin(r^2)}{r^2}[/itex]

[itex]\displaystyle =\lim_{r\,\to\,0^{+}}\frac{d(\sin(r^2))/dr}{d(r^2)/dr}[/itex]​
This is not what you have.
 
Last edited:
  • #5
bubbers said:

Homework Statement



Using polar coordinates, show that lim (x,y)->(0,0) [sin(x^2+y^2)]/[x^2+y^2] = 1

Homework Equations



r^2=x^2+y^2

The Attempt at a Solution



I was able to get the limit into polar coordinates:

lim r->0^+ [sin(r^2)]/r^2

If you let ##\theta=r^2## does the limit become anything you have already studied?
 

1. What are polar coordinates?

Polar coordinates are a coordinate system used to represent points in a two-dimensional plane. They use a distance from the origin and an angle from a reference direction to describe a point.

2. How do polar coordinates relate to the given limit?

In this limit, we are evaluating the function f(x,y) = sin(x^2+y^2)/(x^2+y^2) as the point (x,y) approaches the origin, which can be represented in polar coordinates as (r,θ) with r=√(x^2+y^2) and θ=arctan(y/x). This allows us to rewrite the function as f(r,θ) = sin(r^2)/r^2, which is easier to evaluate.

3. How can I prove the given limit using polar coordinates?

To prove the limit, we can use the squeeze theorem. We can show that the function f(r,θ) = sin(r^2)/r^2 is always between 0 and 1, and as r approaches 0, the function approaches 1. This means that the limit must also be 1.

4. Can this limit be evaluated using other coordinate systems?

Yes, it can also be evaluated using Cartesian coordinates. However, using polar coordinates allows for a simpler and more elegant proof, as shown above.

5. Are polar coordinates commonly used in scientific calculations?

Polar coordinates are commonly used in many fields of science, such as physics, engineering, and mathematics. They are particularly useful in situations involving circular or rotational motion, as well as in solving problems involving symmetry.

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