LIMITS - Multi-variable calculus

In summary, your professor showed that there is a limit to a function but it does not exist if y=x^3.
  • #1
Odyssey
87
0
Hi,

I have an example from my prof. He said, in evaluating limits, we can use L1 (along x-axis), L2 (along y-axis), and L3 (y = kx). After that, if the limits come to the same finite number, then the limit might exist. Here's the example.

[tex]\lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} f(x,y)=\frac{x^3y}{x^6+y^2}[/tex]

The L1, L2, L3 limits all come to zero.

Then being the prof because he is smart, he used L4 = y = x^3, and showed the limit equals some other number, which proved the limit doesn't exist.

[tex]\lim_{\substack{x\rightarrow 0\\y\rightarrow x^3}} f(x,y)=\frac{x^3y}{x^6+y^2}=1/2[/tex]

How can I, as a student, come up with some "random" function such as [itex]y=x^3[/itex] to show that the limit does not exist??

How do I know if the limit exist for sure?? Is there a way to tell??

:confused: :confused: :confused: Thanks for the help! :smile:
 
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  • #2
That's funny. It seems to me the limit does exist and equals 0.

The easiest way to see this is to notice the function is continuous. Then you can just plug in x=y=0.

How on Earth did your prof showed
[tex]\lim_{\substack{x\rightarrow 0\\y\rightarrow x^3}} f(x,y)=\frac{xy}{3+x^2y^2}=1/2[/tex]??
 
  • #3
Your professor is dead wrong.
1) xy is continuous (easily proven)
2) [tex]3+x^{2}y^{2}[/tex] is continuous, and moreover, greater than zero (easily proven)
3) This means that the fraction is also continuous (at all points), and hence, has limits there as well
 
  • #4
Oh crap. I posted the wrong limit! :blushing: :uhh:
My apologies!

[tex]\lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} f(x,y)=\frac{x^3y}{x^6+y^2} = 1/2[/tex]

Sorry again for the carelessness! :blushing: (extended apologoes to Galileo and Arildno)
 
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  • #5
Your professor was right after all, then..:wink:
 
  • #6
Yeah...so how can I know how to plug a function such as y=x^3 to show that the limit DNE??

He only taught us to use the L1, L2, L3 method.
 
  • #7
There isn't a surefire way to solve a limit question (or any problem for that matter).
In this case you just have to notice that the expression y=x^3 will make the two terms in the denominator both of degree 6.
 
  • #8
There is no way of knowing WHAT "counter-example" will work- no matter how many curves you use on which the limit is the same, you can't be sure there isn't some curve on which it is different.

What you COULD do is convert to polar coordinates. That way, the distance to (0,0), which is what is important, is just the single variable r.

In this particular case, You would find that the limit also depends on θ so taking r-> 0 does not give a limit.
 

1. What is a limit in multi-variable calculus?

A limit in multi-variable calculus is the value that a function approaches as its inputs approach a given point, usually denoted as (x,y) or (x,y,z). It is a fundamental concept in calculus that is used to analyze the behavior of functions at specific points.

2. How is a limit in multi-variable calculus different from a limit in single-variable calculus?

A limit in multi-variable calculus is different from a limit in single-variable calculus because it involves considering the behavior of a function in multiple dimensions, rather than just one. This means that the inputs are approaching a point in a multi-dimensional space, rather than on a single number line.

3. What is the purpose of finding limits in multi-variable calculus?

The purpose of finding limits in multi-variable calculus is to understand the behavior of a function at a specific point and to determine whether the function is continuous at that point. This can be useful in solving real-world problems, such as optimizing functions in economics, physics, and engineering.

4. How do you determine if a limit exists in multi-variable calculus?

A limit exists in multi-variable calculus if the outputs of the function approach the same value regardless of the direction in which the inputs approach the given point. This can be determined by evaluating the limit along different paths and comparing the results. If they all approach the same value, then the limit exists.

5. Can a limit in multi-variable calculus be undefined?

Yes, a limit in multi-variable calculus can be undefined if the function has a discontinuity at the given point. This means that the outputs of the function do not approach a single value, regardless of the direction in which the inputs approach the point. In this case, the limit does not exist.

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