Limits if polar coordinates (conceptual explanation)

In summary, polar coordinates are a way of representing points in a two-dimensional space using a distance and angle from a reference axis. They are related to rectangular coordinates, with a point represented by its x and y coordinates in rectangular coordinates and by distance and angle in polar coordinates. In polar coordinates, a limit is a value that a function approaches as the distance from the origin approaches a specific value, and it can be calculated by converting to rectangular coordinates and using standard methods. Limits in polar coordinates have real-world applications in fields such as physics, engineering, and navigation. They can be used to model and analyze situations such as projectile motion, structural design, and navigation on curved surfaces.
  • #1
negation
818
0
1. Homework Statement [/b]

Suppose the lim(x,y) →(0,0) (xy)/SQRT[x^2 + y^2] if it exists

find the limit.

The Attempt at a Solution



x = rcosΘ
y = r sinΘ
r = SQRT[x^2 + y^2]

∴ limr → 0 (r2cosΘrsinΘ)/ r = rcosΘsinΘ [itex]\leq r[/itex]

and so -r [itex]\leq(xy)/SQRT[x^2 + y^2][/itex] [itex]\leq r[/itex]

...
...

I can understand the limit goes to zero because algerbraic multiplication and the sandwhich theorem tells me so. However, the highlighted part in red confuses me. What is the significance of
rcosΘsinΘ [itex]\leqr[/itex]?
 
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  • #2
It was to generate the inequality - they used cosΘrsinΘ ≤ 1.
 

FAQ: Limits if polar coordinates (conceptual explanation)

What exactly are polar coordinates?

Polar coordinates are a way of representing points in a two-dimensional space using a distance (r) from the origin and an angle (θ) measured from a reference axis.

How do polar coordinates relate to rectangular coordinates?

Polar coordinates and rectangular coordinates are two different ways of representing the same points in a two-dimensional space. In rectangular coordinates, a point is represented by its x and y coordinates, while in polar coordinates, it is represented by the distance from the origin and the angle from a reference axis.

What is the concept of a limit in polar coordinates?

In polar coordinates, a limit is a value that a function approaches as the distance from the origin (r) approaches a specific value, often denoted as a. This value can be thought of as the value of the function at that particular point.

How are limits in polar coordinates calculated?

Limits in polar coordinates are calculated by converting the polar coordinates to rectangular coordinates, then applying the standard methods for calculating limits in rectangular coordinates. This may involve techniques such as substitution, factoring, and L'Hôpital's rule.

What are some real-world applications of limits in polar coordinates?

Limits in polar coordinates are used in various fields, such as physics, engineering, and navigation, to model and analyze real-world situations. For example, in physics, limits in polar coordinates can be used to determine the maximum height and range of a projectile. In engineering, limits in polar coordinates can be used to optimize the design of structures, such as bridges and buildings. In navigation, limits in polar coordinates can be used to calculate the shortest distance between two points on a curved surface, such as the Earth's surface.

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