Finding the limit of a multivariable function

In summary, the limit of the expression ##\frac{e^t-1}{t}## as ##\textbf{x}\to\textbf{0}## and ##t\to0## is 1. To show that this limit exists, one needs to prove that it remains the same no matter how the origin is approached. Approaching along ##x_1=0## or ##x_3=0## does not simplify the expression, but approaching in spherical coordinates allows for a simpler limit as ##\rho## goes to 0.
  • #1
schniefen
178
4
Homework Statement
Find the limit of the function below as ##\textbf{x}\to\textbf{0}##.
Relevant Equations
##\frac{e^{|\textbf{x}|^2}-1}{|\textbf{x}|^2+x^2_1x_2+x^2_2x_3}## where ##\textbf{x}=(x_1,x_2,x_3)## and ##|\textbf{x}|=\sqrt{x^2_1+x^2_2+x^2_3}##.
If one approaches the origin from where ##x_2=0##, the terms ##x^2_1x_2+x^2_2x_3## in the denominator equal ##0##. Substituting ##|\textbf{x}|^2## for ##t## yields the expression ##\frac{e^t-1}{t}##, which has limit 1 as ##\textbf{x}\to\textbf{0}## and thus ##t\to0##. So the limit should be 1 if it exists. How could one show that it does exist?
 
  • Like
Likes Delta2
Physics news on Phys.org
  • #2
schniefen said:
How could one show that it does exist?
You need to prove that, no matter how you approach the origin, the limit will always be the same...
Try a few candidates that might yield something different, to get a feeling.
 
  • #3
BvU said:
You need to prove that, no matter how you approach the origin, the limit will always be the same...
Try a few candidates that might yield something different, to get a feeling.

Yes, but approaching along ##x_1=0## or ##x_3=0## doesn't simplify the expression, does it?
 
  • #4
Oh, doesn't it :rolleyes: ?
What comes out ?
 
  • #5
It only cancels one of the terms in the sum ##x^2_1x_2+x^2_2x_3## for ##x_1=0## and ##x_3=0## respectively.
 
  • #6
And what is left over ?

Tip: substitute ## | x| = \sqrt {...} ## to get a clearer picture, especially in the denominator 😉 .
 
  • Like
Likes schniefen
  • #7
Bedtime here (UTC+2) :sleep:
 
  • #8
With a multi-variable limit, [itex]\left(x_1, x_2, x_3\right)[/itex] going to (0, 0, 0), it might be best to convert to spherical coordinates. That way the limit is just as [itex]\rho[/itex] goes to 0. If that limit depends on [itex]\theta[/itex] or [itex]\phi[/itex] there is no limit. If it is a single number, then that is the limit.
 
  • Like
Likes schniefen and Delta2

1. What is a multivariable function?

A multivariable function is a mathematical function that depends on more than one independent variable. It can be written in the form f(x, y) or z = f(x, y), where x and y are the independent variables and z is the dependent variable.

2. Why is it important to find the limit of a multivariable function?

Finding the limit of a multivariable function is important because it helps us understand the behavior of the function as the independent variables approach a certain value. It also allows us to determine if the function is continuous at a certain point, which is essential in many real-world applications.

3. How is the limit of a multivariable function calculated?

The limit of a multivariable function is calculated by evaluating the function at different points that approach the desired value of the independent variables. This can be done by plugging in the values of the independent variables and observing the resulting values of the function.

4. What are some common techniques for finding the limit of a multivariable function?

Some common techniques for finding the limit of a multivariable function include direct substitution, factoring, and using trigonometric identities. Other methods such as L'Hôpital's rule and Taylor series expansion can also be used for more complex functions.

5. Are there any limitations to finding the limit of a multivariable function?

Yes, there are limitations to finding the limit of a multivariable function. In some cases, the limit may not exist or may be different depending on the path taken to approach the desired value of the independent variables. Additionally, some functions may be too complex to find the limit analytically and may require the use of numerical methods.

Similar threads

  • Calculus and Beyond Homework Help
Replies
10
Views
819
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
10
Views
903
  • Calculus and Beyond Homework Help
Replies
12
Views
760
  • Calculus and Beyond Homework Help
Replies
4
Views
898
  • Calculus and Beyond Homework Help
Replies
8
Views
648
Replies
9
Views
699
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
1K
Back
Top