# Using polar coordinates to evaluate a multivariable limit

• Jimmy25
In summary, when evaluating a multivariable limit using polar coordinates, it is important to consider the dependence on the path to the point and not treat theta as a constant.
Jimmy25

## Homework Statement

When you substitute polar coordinates into a multivariable limit, do you treat theda as a constant when evaluating? (I know how to use polar coordinates to evaluate a limit but haven't learned what they are yet)

Jimmy25 said:

## Homework Statement

When you substitute polar coordinates into a multivariable limit, do you treat theda as a constant when evaluating? (I know how to use polar coordinates to evaluate a limit but haven't learned what they are yet)

Generally you wouldn't treat $$\theta$$ as a constant because the most important concept in taking a multivariable limit is that the limit doesn't exist if it depends on the path you take to the point.

As an example, consider

$$\lim_{(x,y)\rightarrow (0,0)} \frac{x^2y}{x^4+y^2} = \lim_{r->0} \frac{r\cos^2\theta\sin\theta}{r^2 \cos^4\theta + \sin^2\theta} = \lim_{r->0} \frac{r \cos^2\theta}{\sin\theta}.$$

If we just take $$r\rightarrow 0$$, we find that this vanishes. However, if we also take $$\theta\rightarrow 0$$ at the same rate as r, we find $$\cos^2(0)=1$$. Therefore the limit does not exist.

## 1. What are polar coordinates?

Polar coordinates are a coordinate system used to locate points in a plane using a distance from the origin (called the radius) and an angle from a fixed reference line (usually the positive x-axis).

## 2. How do you convert from rectangular coordinates to polar coordinates?

To convert from rectangular coordinates (x,y) to polar coordinates (r,θ), you can use the following equations:
r = √(x²+y²)
θ = tan⁻¹(y/x)

## 3. How do you evaluate a multivariable limit using polar coordinates?

To evaluate a multivariable limit using polar coordinates, you can substitute the polar coordinates (r,θ) into the given function and simplify. Then, you can take the limit as r approaches 0 to determine the overall limit.

## 4. Why is it sometimes easier to use polar coordinates when evaluating a multivariable limit?

Polar coordinates can be useful when evaluating a multivariable limit because they can simplify the expression and eliminate any complicated algebraic terms. Additionally, polar coordinates can often make it easier to visualize the limit and understand the behavior of the function.

## 5. Are there any limitations to using polar coordinates to evaluate multivariable limits?

Yes, there are some limitations to using polar coordinates to evaluate multivariable limits. In some cases, the function may not be defined at the origin or the limit may not exist even though the polar form of the function approaches a finite value. Additionally, some functions may have more complicated polar representations, making it difficult to evaluate the limit using this method.

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