Is the Limit of x^4 and y^4 as (x,y)->(0,1) Equal?

In summary, the conversation discusses the concept of limits in calculus and how they can be affected by approaching a point from different paths. The speaker initially believes the limit at (0,1) does not exist due to obtaining different values when approaching from different paths, but realizes their reasoning is false when considering a different substitution. The conversation also references a "war" between engineers and scientists regarding the concept of limits.
  • #1
ImAnEngineer
209
1
In my calculus book it says that the limit of y(x^3) as (x,y)->(0,1) equals 0. It also says that a limit does not exist if you obtain different values when approaching (0,1) from different paths.
It is easy to see the limit is zero by using the product rule for limits. However, if I set x=y, we get the limit of x^4 as (x,y)->(0,1) or the limit of y^4 as (x,y)->(0,1) which are clearly not equal. Hence the limit does not exist (?).

Where is my reasoning false?
 
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  • #2
ImAnEngineer said:
In my calculus book it says that the limit of y(x^3) as (x,y)->(0,1) equals 0. It also says that a limit does not exist if you obtain different values when approaching (0,1) from different paths.
It is easy to see the limit is zero by using the product rule for limits. However, if I set x=y, we get the limit of x^4 as (x,y)->(0,1) or the limit of y^4 as (x,y)->(0,1) which are clearly not equal. Hence the limit does not exist (?).

Where is my reasoning false?

By thinking that you can get to (0,1) along a path where x=y?

Cheers -- sylas
 
  • #3
Haha, how stupid is that! :)

So I could substitute x=y-1, then the limit becomes (y-1)y^4 as (x,y)->(0,1) which DOES equal zero and everything works out. Thanks!
 
  • #4
ImAnEngineer said:
Haha, how stupid is that! :)

So I could substitute x=y-1, then the limit becomes (y-1)y^4 as (x,y)->(0,1) which DOES equal zero and everything works out. Thanks!

You're welcome. Always glad to contribute another data point on the war between engineers and scientists. :biggrin:
 

FAQ: Is the Limit of x^4 and y^4 as (x,y)->(0,1) Equal?

What is a limit in terms of two variables?

A limit in terms of two variables is a mathematical concept that describes the behavior of a function as both of its variables approach a specific value or point. It is used to determine the value that a function approaches as its inputs get closer and closer to a particular point.

How is a limit of a function with two variables calculated?

A limit of a function with two variables is calculated by evaluating the function at various points that approach a specific point. The closer these points get to the specific point, the more accurate the limit becomes. This process is known as taking the limit.

What is the purpose of finding a limit of a function with two variables?

The purpose of finding a limit of a function with two variables is to better understand the behavior of the function as its inputs approach a specific point. It can also be used to determine the continuity and differentiability of a function.

What are some common methods for finding a limit of a function with two variables?

Some common methods for finding a limit of a function with two variables include substitution, factoring, and using algebraic manipulation techniques. Other methods include using graphical representations and numerical approaches such as tables and calculators.

Can a limit of a function with two variables exist at a point where the function is undefined?

Yes, a limit of a function with two variables can exist at a point where the function is undefined. This is because the limit is concerned with the behavior of the function as its inputs approach a specific point, not necessarily the value of the function at that point. However, if the limit does not exist, it can indicate a discontinuity or a jump in the function at that point.

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