Solving Limit Using Polar Coordinates

In summary: The first thing to do is to change your expression to polar coordinates.According to the rules of this Forum, you need to show some effort before we can help you.I've done it, but do not know how to continue. I wanted to see the resolution step by step, with comments about what is being done, of course if you can.We will do no such thing.
  • #1
Tiago1818
6
0
Staff was trying to understand a matter of calculation. I hope someone can explain me in detail how to solve this limit using polar coordinates:

http://img36.imageshack.us/img36/2667/semttulokej.png
 
Last edited by a moderator:
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  • #2
Tiago1818 said:
Staff was trying to understand a matter of calculation. I hope someone can explain me in detail how to solve this limit using polar coordinates:

<img src="http://latex.codecogs.com/gif.latex?\lim_{(x,y)\to(0,0)} \frac{sen(x^2 + y^2)}{1-cos\sqrt[]{x^2 + y^2}}" title="\lim_{(x,y)\to(0,0)} \frac{sen(x^2 + y^2)}{1-cos\sqrt[]{x^2 + y^2}}" />
Hello Tiago1818. Welcome to PF !

Here's an attempt to reconstruct what was in your image.

[itex] \displaystyle \lim_{(x,y)\to(0,0)}\ \ \frac{\sin(x^2 + y^2)}{1-\cos\sqrt{x^2 + y^2}}[/itex]

The first thing to do is to change your expression to polar coordinates.

According to the rules of this Forum, you need to show some effort before we can help you.
 
Last edited:
  • #3
I've done it, but do not know how to continue. I wanted to see the resolution step by step, with comments about what is being done, of course if you can.I am Brazilian and I'm using google translator, so what I am saying may seem a bit confusing.
 
  • #4
Tiago1818 said:
I wanted to see the resolution step by step, with comments about what is being done, of course if you can.

We will do no such thing.

YOU need to show what you tried first, then we will GUIDE you in the right direction. We do not simply give out the answers.
 
  • #5
I did it.

http://img825.imageshack.us/img825/237/digitalizar0029.png

I do not know what to do now.
 
Last edited by a moderator:
  • #6
know yet, the answer is "2"
 
  • #8
Hopefully, you know the limit: [itex]\displaystyle \lim_{x\to0}\ \frac{\sin(x)}{x}\,.[/itex]

Multiply the numerator & denominator by 1 + cos(r) .
 
  • #9
I did so, using L'Hospital. He had no knowledge of this theorem. Thanks for the tip.

http://img6.imageshack.us/img6/4122/digitalizar0030b.jpg
 
Last edited by a moderator:
  • #10
Two of the very first trig limits that most people learn in Calculus are
[tex]\lim_{x\to 0}\frac{sin(x)}{x}= 1[/tex]
and
[tex]\lim_{x\to 0}\frac{1- cos(x)}{x}= 0[/tex]
 

1. What are polar coordinates and how are they used to solve limits?

Polar coordinates are a coordinate system used to locate a point in a two-dimensional plane. They consist of a distance from the origin (the pole) and an angle from a reference line (usually the positive x-axis). To solve a limit using polar coordinates, we convert the given Cartesian coordinates into polar coordinates and then evaluate the limit as the distance from the origin approaches zero.

2. Why do we use polar coordinates to solve limits?

Polar coordinates can be useful when dealing with limits involving circular or symmetric functions. In these cases, it may be easier to convert the Cartesian coordinates into polar coordinates and then evaluate the limit. Additionally, polar coordinates can help simplify the algebraic manipulation of equations, making it easier to evaluate limits.

3. Can all limits be solved using polar coordinates?

No, not all limits can be solved using polar coordinates. Polar coordinates are most useful when dealing with limits involving circular or symmetric functions. If the function is not circular or symmetric, it may be more appropriate to use another method, such as direct substitution or L'Hopital's rule.

4. Are there any limitations to using polar coordinates to solve limits?

One limitation of using polar coordinates is that not all functions can be easily expressed in terms of polar coordinates. In some cases, it may be difficult or impossible to convert the given Cartesian coordinates into polar coordinates. Additionally, polar coordinates may not be the most efficient method for solving certain types of limits.

5. Can we use polar coordinates to solve limits in higher dimensions?

Yes, polar coordinates can be extended to higher dimensions, such as three-dimensional space. In these cases, we would use spherical coordinates, which consist of a distance from the origin, an angle from a reference plane, and an angle from a reference line. The same principles of converting coordinates and evaluating limits as the distance approaches zero still apply in higher dimensions.

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