Limit Evaluation: (x^4+y^4)/((x^2+y^2)^(3/2)) as (x,y) -> (0,0)

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In summary, limit evaluation is a process used in calculus to determine the behavior of a function as the input approaches a certain value or point. The limit of (x^4+y^4)/((x^2+y^2)^(3/2)) as (x,y) approaches (0,0) is 0 and can be evaluated using the limit laws and the fact that the square root function is continuous. If the limit of a function is undefined, it means that the function does not approach a single value as the input approaches a certain point. Limit evaluation is used in various fields, such as physics, engineering, economics, and computer science.
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kasse
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Evaluate the liit of (x^4+y^4)/((x^2+y^2)^(3/2)) when (x,y) approaches (0,0)

I'm trying to replace x and y with rcost and ycost, but it seems too complex. How can I abbreviate the equation?
...

I solved it myself...The limit is 0.
 
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  • #2
No need to abbreviate, just analyse the r dependence,
and check that the angular dependence is a has no singularity.

And indeed you are right: 0 is the limit.
 

1. What is limit evaluation?

Limit evaluation is a process used in calculus to determine the behavior of a function as the input approaches a certain value or point.

2. What is the limit of (x^4+y^4)/((x^2+y^2)^(3/2)) as (x,y) approaches (0,0)?

The limit of (x^4+y^4)/((x^2+y^2)^(3/2)) as (x,y) approaches (0,0) is 0. This means that as (x,y) gets closer and closer to (0,0), the value of the function approaches 0.

3. How do you evaluate the limit of (x^4+y^4)/((x^2+y^2)^(3/2)) as (x,y) approaches (0,0)?

To evaluate this limit, we can use the limit laws and rewrite the function as (x^2+y^2)^2/((x^2+y^2)^(3/2)). Then, we can simplify by canceling out the (x^2+y^2) terms and we are left with the limit of (x^2+y^2)^(1/2) as (x,y) approaches (0,0). Using the fact that the square root function is continuous, we can evaluate this limit as 0.

4. What does it mean if the limit of a function is undefined?

If the limit of a function is undefined, it means that the function does not approach a single value as the input approaches a certain point. This could be due to a variety of reasons, such as a jump or discontinuity in the function, or the function approaching different values from different directions.

5. How is limit evaluation used in real life?

Limit evaluation is used in real life in many fields, including physics, engineering, and economics. For example, in physics, limit evaluation is used to calculate instantaneous velocity and acceleration. In economics, it is used to determine the marginal cost and revenue of a product. It is also used in computer science to calculate the time complexity of algorithms.

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