Finding the Limit of a Multivariable Function at (0,0)

In summary, the limit of a function is the value that the function approaches as the input value gets closer and closer to a specific value. This is different from the value of a function at a specific point, which is the output value of the function at that point. The formal definition of continuity states that a function is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point. There are three types of discontinuities: removable, jump, and infinite. Limits can be used to evaluate the continuity of a function by determining if the limit at a specific point exists and is equal to the function's value at that point. If they are equal, the function is continuous
  • #1
ahhppull
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Homework Statement


lim of (y^2)(sin^2x) /(x^4+y^4) as (x,y) approaches (0,0)


Homework Equations





The Attempt at a Solution



I got the limit as (x,y) approaches (0,y) and as (x,y) approaches (x,0), and it equals 0. But now I'm unsure of what to to next. I think it was the limit as (x,y) approaches (x,x) when x=y, but i get sin^2x / 2x^2
 
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  • #2
[tex]\lim_{x\to 0}\frac{\sin^2x}{2x^2}=\frac{1}{2}\lim_{x\to 0}\left(\frac{\sin x}{x}\right)^2=\frac{1}{2}[/tex]
 

FAQ: Finding the Limit of a Multivariable Function at (0,0)

What is the definition of limit?

The limit of a function is the value that the function approaches as the input value gets closer and closer to a specific value.

How is a limit different from the value of a function at a specific point?

A limit represents the behavior of a function as the input value gets closer to a specific value, while the value of a function at a specific point is the output value of the function at that point.

What is the formal definition of continuity?

A function is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point.

What are the different types of discontinuities?

There are three types of discontinuities: removable, jump, and infinite. Removable discontinuities occur when the limit and value of the function at a point are not equal, but can be made equal by redefining the function at that point. Jump discontinuities occur when the limit of the function at a point exists, but the value of the function at that point does not. Infinite discontinuities occur when the limit of the function at a point does not exist because it approaches infinity.

How can limits be used to evaluate the continuity of a function?

If the limit of a function at a point exists and is equal to the value of the function at that point, then the function is continuous at that point. However, if the limit does not exist or is not equal to the value of the function, then the function is not continuous at that point.

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