Need Help with Multivariable Limit? Find Solutions Here!

In summary, the conversation discusses finding the limit of x^(-y) as (x,y) approaches (1,oo). The speaker mentions trying to switch to polar coordinates, but is unsure if it proves anything. They then suggest trying some examples, such as xn=(1+1/n) and yn=n, and xn=(1+2/n). The other person in the conversation suggests considering the sequence 1/e, 1/e^2, 1/e^3... as an alternative approach.
  • #1
rman144
35
0
I've been stuck on this problem for quite a while now and could use some assistance:

Find the limit (or prove that it does not exist):

lim{(x,y)->(1^+,oo)} x^(-y)


I've tried switching to polar and end up with y=rsin(@) implying r diverges, which implies cos(@) must tend to zero for x to approach 1, but I'm not certain this actually proves or disproves anything. Honestly, any help would be much appreciated.
 
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  • #2
Try some examples. Take xn=(1+1/n) and yn=n and let n go to infinity. Then xn->1+ and yn->infinity. What's the limit of xn^(-yn)? Then try xn=(1+2/n). Conclusion?
 
  • #3
Why didn't I think of that? 1/e, 1/e^2, 1/e^3...

Lol, thanks.
 

FAQ: Need Help with Multivariable Limit? Find Solutions Here!

What is a multivariable limit?

A multivariable limit is a mathematical concept used to describe the behavior of a function as multiple variables approach a certain point. It is similar to a regular limit, but takes into account the effect of multiple variables on the function's output.

How is a multivariable limit calculated?

To calculate a multivariable limit, you must first isolate the variables that are approaching the point of interest. Then, you can substitute different values for those variables and observe the output of the function. The limit is the value that the function approaches as the variables get closer and closer to the point.

What is the significance of multivariable limits in science?

Multivariable limits are important in science because they can help us understand the behavior of complex systems. By studying the limits of functions with multiple variables, we can gain insights into the behavior of physical systems, such as fluid flow, electric fields, and chemical reactions.

What are some common techniques for evaluating multivariable limits?

Some common techniques for evaluating multivariable limits include using algebraic manipulation, substitution, and graphing. These techniques can help us understand the behavior of a function as multiple variables approach a given point.

How can multivariable limits be used to solve real-world problems?

Multivariable limits can be used to solve real-world problems by helping us model and predict the behavior of complex systems. For example, in engineering, multivariable limits can be used to optimize the design of structures or predict the behavior of materials under different conditions.

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