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

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Discussion Overview

The discussion centers on the limit of the expression involving x and y as they approach the point (0,1). Participants explore the conditions under which the limit exists and the implications of approaching the limit along different paths.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant references a calculus book stating that the limit of y(x^3) as (x,y) approaches (0,1) equals 0, but questions the existence of the limit when approaching along the path where x=y, leading to different values.
  • Another participant suggests that substituting x=y-1 simplifies the limit to (y-1)y^4 as (x,y) approaches (0,1), which they claim equals zero.
  • There is a humorous acknowledgment of the complexity of the problem and the ongoing debate between engineers and scientists regarding limits.

Areas of Agreement / Disagreement

Participants express differing views on the existence of the limit, with some arguing it does not exist due to path dependence, while others propose alternative substitutions that yield a limit of zero. The discussion remains unresolved regarding the overall existence of the limit.

Contextual Notes

Participants rely on different paths to approach the limit, which leads to varying conclusions. There is an implicit assumption that the limit's existence is contingent on the path taken, but this is not universally accepted among participants.

ImAnEngineer
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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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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
 
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!
 
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:
 

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