Lim x,y→0: Multivariable Limit

In summary, the task is to find the limit as x and y approach 0 of (10sin(x^2 + y^2)) / (x^2 + y^2). Direct substitution and multiplying by the conjugate result in an indeterminate form, so approaching (0,0) along different paths such as (0,x) and (x,0) is suggested. The path (t,t) seems promising, and plugging in x=t and y=t gives the same indeterminate form. To find a way around this, L'Hospital's rule is proposed, but it is mentioned that it does not apply in three dimensions. However, since the squared distance r^2 goes to 0 as (x,y
  • #1
nate9519
47
0

Homework Statement


find lim as x,y approach 0 of (10sin(x^2 + y^2)) / (x^2 + y^2)

Homework Equations

The Attempt at a Solution


direct substitution yields indeterminate form and so does multiplying by the conjugate. what other methods are there to use?
 
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  • #2
So you want to find:

$$\displaystyle \lim_{(x,y) \rightarrow (0,0)} \frac{10sin(x^2 + y^2)}{x^2 + y^2}$$

Try approaching ##(0, 0)## along different paths to zero such as ##(0, x)## and ##(x, 0)##.

EDIT: Path ##(t, t)## looks promising.
 
  • #3
for the paths (0,x) and (x,0) I got 10sin(y^2) / y^2. does that mean the limit is 10sin(y^2) / y^2
 
  • #4
nate9519 said:
for the paths (0,x) and (x,0) I got 10sin(y^2) / y^2. does that mean the limit is 10sin(y^2) / y^2

What is:

$$\displaystyle \lim_{x \rightarrow 0} \frac{10sin(x^2)}{x^2}$$

Looks like a first year problem.
 
  • #5
wow. can't believe I didn't see that . so that is indeterminate but I know the limit exists because the problem says "Hint - the limit does exist". when you said the path (t,t) looked promising I assumed you meant parameterizing x and y. but what do I let them equal
 
  • #6
Indeed you can approach ##(0, 0)## along several different paths and get the same answer. That's how you know the limit exists and is finite. Plugging in ##x = t## and ##y = t## will give the same limit.
 
  • #7
so the paths (x,0) (0,x) (y,0) (0,y) and (t,t) all give indeterminate forms. I am just not seeing a way around this
 
  • #8
nate9519 said:
so the paths (x,0) (0,x) (y,0) (0,y) and (t,t) all give indeterminate forms. I am just not seeing a way around this

Why not apply L'Hospital's rule? If the form is ##0/0## you can easily find the limit that way.

Although the conventional way would be to recognize: ##\displaystyle \lim_{x \rightarrow 0} \frac{sin(x)}{x} = 1##
 
  • #9
I was told l'hospital's rule did not apply in three dimensions. but since one variable goes away when evaluated on (x,0) , (y,0), etc... does that mean its like that variable never existed
 
  • #10
nate9519 said:
I was told l'hospital's rule did not apply in three dimensions. but since one variable goes away when evaluated on (x,0) , (y,0), etc... does that mean its like that variable never existed

No matter how ##(x,y) \to (0,0)## the distance of ##(x,y)## from the origin goes to 0. In other words, the squared distance ##r^2 = x^2 + y^2 \to 0##. In fact, ##r^2 \to 0## if, and only if ##(x,y) \to (0,0)## in some way. Now go back and re-examine your original function ##f(x,y)##.
 
  • #11
nate9519 said:
I was told l'hospital's rule did not apply in three dimensions. but since one variable goes away when evaluated on (x,0) , (y,0), etc., does that mean it's like that variable never existed?
Yes. The limits you end up with using those paths only depend on one variable, so you can use the Hospital rule on them. Unfortunately, all you'll have shown is that the limit along those paths exist. You actually need to show the original limit exists for all possible paths to the origin. You want to think about what Ray said.
 

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 a way of analyzing how a function changes when more than one input variable is changed simultaneously.

How is a multivariable limit different from a single variable limit?

In a single variable limit, only one input variable is changing while the others remain constant. In a multivariable limit, multiple input variables are changing at the same time, making it a more complex concept to understand and calculate.

What is the purpose of calculating a multivariable limit?

Calculating a multivariable limit can help us understand the behavior of a function in higher dimensions. It can also help us determine if a function is continuous at a particular point, and can be used in optimization problems.

How do you calculate a multivariable limit?

To calculate a multivariable limit, you must approach the point of interest along different paths and see if the function approaches the same value. If it does, then the limit exists. If it does not, then the limit does not exist.

What are some common types of multivariable limits?

Some common types of multivariable limits include limits where both variables approach the point of interest along the same path, limits where the variables approach the point from different directions, and limits that involve polar or spherical coordinates.

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