Multivariable Calculus Limits

In summary, the conversation discusses finding the limit of a function as (x,y) approaches (0,0). Different techniques are suggested, including using polar coordinates and L'Hopital's rule. The final conclusion is that the limit does not exist due to the function approaching infinity as the distance to (0,0) decreases.
  • #1
eutopia
28
0
lim of

cos((x^2 + y^2) - 1)/(x^2 + y^2)

as (x,y) approaches (0,0)

I have no clue how to tackle this problem. I tried to find the level set so at least I can have a clue of what the graph looks like, but then, I didn't know how to find the level sets either. If I set c = the equation, I have 2 unknowns so I cannot solve, and its not an obvious graph like a circle or something. On the other hand, I tried l'hopitale but that needs the derivative and what in the world am i taking a derivative in terms of since there are 2 variables?

I'm very confused. PLEASE HELP! :bugeye:
 
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  • #2
you have an intermediate form of lahopital's theorey of the form 1/0
do you know how to do these?
 
  • #3
I think he's missing a parenthesis -- the numerator is supposed to be cos(x²+y²) - 1. (P.S. 1/0 is not indeterminate, and AFAIK it's not L'Hôpital theory)

eutopia: what techniques have you seen used for similar problems? There is one in particular that makes this problem very simple.


its not an obvious graph like a circle or something.

Are you sure about that?
 
  • #4
For the limit to exist it has to exist regardless of the direction from which you are approaching the point. Parameterize lines passing through the origin and see if you can get that.
 
  • #5
As you said, the limit has to exist (and be the same value) for any way you approach the origin -- just looking at the lines isn't good enough.
 
  • #6
"Lines through the origin", suggested by MalleusScientiarum, will help show that a limit does not exist by getting, hopefully, different limits on different lines. But they can't prove that a limit DOES exit (or find it) since even if the limit is the same along all lines, there might be other curves, not lines, passing through the origin that give a different limit.

The best way to handle ANY limit problem in more than one variable (going to (0,0) or (0,0,0), etc.) is to change to polar (spherical, etc.) coordinates since that way one variable, r (ρ, etc.) measures the distance to (0,0) directly! In this case, that's easy since x and y only appear in x2+ y2= r2.

The original function,
[tex]\frac{cos((x^2 + y^2) - 1)}{x^2 + y^2}[/tex]
becomes
[tex]\frac{cos(r^2-1)}{r^2}[/tex]
which clearly goes to infinity as r goes to 0.

Hurkyls suggested correction,
[tex]\frac{cos(x^2+y^2)-1}{x^2+y^2}[/tex]
becomes
[tex]\frac{cos(r^2)-1}{r^2}[/tex]
which now has only one variable and can be done by L'Hopital's rule. (The limit is 0.)
 

1. What is a multivariable calculus limit?

A multivariable calculus limit is a mathematical concept that describes the behavior of a function as one or more of its variables approach a specific value. It is used to understand how a function changes when its input values change, and is an important tool in analyzing the properties of functions in multiple dimensions.

2. How is a multivariable calculus limit different from a single variable limit?

A multivariable calculus limit involves more than one independent variable, while a single variable limit only has one independent variable. This means that in multivariable calculus, the function can change in multiple directions as the input values change, whereas in single variable calculus, the function can only change along one axis.

3. What is the importance of multivariable calculus limits in real-world applications?

Multivariable calculus limits are important in many areas of science and engineering, such as physics, economics, and computer graphics. They allow us to model and understand complex systems that involve multiple variables, and are essential in solving optimization problems and making predictions about the behavior of these systems.

4. How do you evaluate a multivariable calculus limit?

To evaluate a multivariable calculus limit, you need to approach the specified point from all possible directions and determine if the function approaches a finite value or diverges. This can be done by considering limits along specific paths, using algebraic techniques, or using advanced tools such as L'Hôpital's rule.

5. Are there any special cases when evaluating multivariable calculus limits?

Yes, there are some special cases when evaluating multivariable calculus limits. One common case is when the function is undefined at the specified point, in which case you may need to use substitution or other techniques to evaluate the limit. Another case is when the function has multiple paths that approach the same point, in which case you may need to use different techniques to evaluate the limit along each path.

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