# Multivariable limits (NOT TO THE ORIGIN)

• mirajshah
In summary: It's been a tough week for all of us. In summary, the conversation discusses a problem with finding the limit of a multivariable function at a specific point. The participants consider different approaches and eventually determine that the limit approaches -∞. They also mention difficulties in visualizing the problem and difficulties in using Wolfram Alpha to solve it.
mirajshah

## Homework Statement

Hi everyone! I'm pretty good with multivariable limits, but this one has me stumped:
Find the limit or show that it does not exist:
$$\underset{\left(x,y,z\right)\rightarrow\left(1,-1,1\right)}{\lim}\frac{yz+xz+xy}{1+xyz}$$

## The Attempt at a Solution

I could not work with polar coordinates here because there is no easy way to find a value that ρ approaches (the point is (1, -1, 1).
I found it difficult to prove that the limit didn't exist, as in this case the point is in 4-d space and one can only approach it from various 3-D spaces, which I simply could not visualize how to do.

Last edited:
simply take the limit along one axis, say x axis, the limit already blows up

Read the problem buddy. The limit is to be taken approaching the point (1, -1, 1), which is a point that doesn't lie on any of the axes.

The numerator approaches 0 and the denominator doesn't approach 0. Isn't that enough to tell you about the behavior of the limit?

Doesn't the denominator approach: $\left(1+\left(1\times-1\times1\right)\right)=\left(1+\left(-1\right)\right)=\left(1-1\right)=0$?

sunjin09, I realize I was a little rude with my response and I'm sorry, it's been a tough week. Thanks for the responses guys! I appreciate it.

Quick note: plugging into Wolfram Alpha yields nothing, so I don't have an answer we can cross-reference against. Sorry!

Yes, the denominator approaches 0. And the numerator doesn't. That was Dick's point.

HallsofIvy said:
Yes, the denominator approaches 0. And the numerator doesn't. That was Dick's point.

Yes, I had them backwards, sorry.

Oh my god, the limit is -∞? I'm so sorry guys, I feel like a real idiot. Thanks for the help!

mirajshah said:
Oh my god, the limit is -∞? I'm so sorry guys, I feel like a real idiot. Thanks for the help!

You are welcome! But I wouldn't describe it that way. The denominator doesn't have a definite sign. It could be either +∞ or -∞ depending on how you approach it.

mirajshah said:
Doesn't the denominator approach: $\left(1+\left(1\times-1\times1\right)\right)=\left(1+\left(-1\right)\right)=\left(1-1\right)=0$?

sunjin09, I realize I was a little rude with my response and I'm sorry, it's been a tough week. Thanks for the responses guys! I appreciate it.

Quick note: plugging into Wolfram Alpha yields nothing, so I don't have an answer we can cross-reference against. Sorry!

No worries. I should've been more accurate in my wording.

## What is a multivariable limit?

A multivariable limit is a mathematical concept that describes the behavior of a function as its input variables approach a specific point or value. It is a fundamental concept in multivariable calculus and is used to analyze the behavior of functions in multiple dimensions.

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

A multivariable limit involves functions with more than one input variable, while a single variable limit involves functions with only one input variable. In a multivariable limit, the input variables approach a specific point in a multidimensional space, while in a single variable limit, the input variable approaches a specific point on a one-dimensional number line.

## What is the notation used for multivariable limits?

The notation used for multivariable limits is similar to that used for single variable limits, but with the addition of subscripts to denote the specific variables involved. For example, the notation for the limit of a function f(x, y) as x and y approach a point (a, b) would be written as lim(x,y)→(a,b)f(x,y).

## How are multivariable limits evaluated?

Multivariable limits are evaluated by approaching the given point or value along different paths or directions. This is because the behavior of a function may vary depending on the direction in which the input variables approach the point. If the limit exists, it will be the same regardless of the path taken.

## What are some real-world applications of multivariable limits?

Multivariable limits have many real-world applications, particularly in physics and engineering. They are used to analyze the behavior of physical systems, such as the flow of fluids or the motion of objects in space. They are also used in optimization problems, where the goal is to find the maximum or minimum value of a function with multiple variables.

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