Does y*ln(y^2+x^2) Approach Zero as (x, y) Goes to (0,0)?

In summary, the conversation discusses finding the limit of the function (x,y) --> (0,0) of y*ln(y^2+x^2), using polar coordinates and various paths. The conclusion is that the limit is equal to 0.
  • #1
AHMAD MS
1
0

Homework Statement



(x,y)-->(0,0) y*ln (y2+x2)

Homework Equations



!

The Attempt at a Solution


i've tried polar coordinates i end up with sinθ in it don't know how to proceed

and I've taken several paths like y=0 , x=0 , x=y2 , x=y , x=y3
they all yeild zero

Edit : is it right if i after making polar sub. to say that :

r*sinθ * ln r2 [itex]\leq[/itex] r*ln r2
because if its right then problem solved :P
 
Last edited:
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  • #2
AHMAD MS said:

Homework Statement



(x,y)-->(0,0) y*ln (y2+x2)

Homework Equations



!

The Attempt at a Solution


i've tried polar coordinates i end up with sinθ in it don't know how to proceed

and I've taken several paths like y=0 , x=0 , x=y2 , x=y , x=y3
they all yeild zero

Edit : is it right if i after making polar sub. to say that :

r*sinθ * ln r2 [itex]\leq[/itex] r*ln r2
because if its right then problem solved :P

Almost. You need absolute values:

|r*sinθ * ln r2| [itex]\leq[/itex] r*|ln (r2)|
 

FAQ: Does y*ln(y^2+x^2) Approach Zero as (x, y) Goes to (0,0)?

What is the limit of a multi-variable function?

The limit of a multi-variable function is the value that the function approaches as the input variables get closer and closer to a specified point. It is denoted by the symbol "lim" followed by the function and the point at which the limit is being evaluated.

How do you calculate the limit of a multi-variable function?

The limit of a multi-variable function can be calculated by evaluating the function at different points close to the specified point and observing the trend of the output values. If the output values approach a certain value as the input values get closer to the specified point, then that value is the limit of the function.

Does the limit of a multi-variable function always exist?

No, the limit of a multi-variable function may not always exist. It may not exist if the function approaches different values from different directions or if it approaches infinity at the specified point.

What is the relationship between continuity and the limit of a multi-variable function?

A multi-variable function is continuous at a point if and only if the limit of the function at that point exists and is equal to the value of the function at that point. In other words, continuity of a function implies the existence of its limit at that point.

Can the limit of a multi-variable function be different at different points?

Yes, the limit of a multi-variable function can be different at different points. This can happen if the function is not continuous at a certain point or if the function is defined differently at different points.

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