Limit of a two-variable function

• Mr Davis 97
In summary, the conversation discusses evaluating the limit of a given function and whether it exists. The attempt at a solution involves trying several curves and converting to polar form, which leads to a new expression. However, there is uncertainty about how to proceed with the simplification.
Mr Davis 97

Homework Statement

Does the following limit exist: ##\displaystyle \lim_{(x,y) \rightarrow (0,0)} = \frac{\sqrt{x^2+y^2+xy^2}}{\sqrt{x^2+y^2}}##?

The Attempt at a Solution

So I am trying to evaluate the limit along several curves, such as y=x, y=0, y=x^2, and I keep getting 1. I can't think of any other curves to evaluate on. Does the limit exist or is there another curve that I am not evaluating that gives a different limit?

Mr Davis 97 said:
So I am trying to evaluate the limit along several curves, such as y=x, y=0, y=x^2, and I keep getting 1
If it does exist, it does not matter how many curves you try you will not have proved it. So how about trying to prove it exists?

Mr Davis 97 said:

Homework Statement

Does the following limit exist: ##\displaystyle \lim_{(x,y) \rightarrow (0,0)} = \frac{\sqrt{x^2+y^2+xy^2}}{\sqrt{x^2+y^2}}##?

The Attempt at a Solution

So I am trying to evaluate the limit along several curves, such as y=x, y=0, y=x^2, and I keep getting 1. I can't think of any other curves to evaluate on. Does the limit exist or is there another curve that I am not evaluating that gives a different limit?
Converting to polar form is fruitful, showing that the limit does exist.

Mr Davis 97
Mark44 said:
Converting to polar form is fruitful, showing that the limit does exist.
Converting to polar form, I get, after simplification, ##\displaystyle \lim_{r \rightarrow 0^+} \sqrt{1+\cos \theta \sin^2 \theta}##. I'm not sure how to proceed

Mr Davis 97 said:
Converting to polar form, I get, after simplification, ##\displaystyle \lim_{r \rightarrow 0^+} \sqrt{1+\cos \theta \sin^2 \theta}##. I'm not sure how to proceed
You seem to have dropped an r.

Mr Davis 97
Mr Davis 97 said:
Converting to polar form, I get, after simplification, ##\displaystyle \lim_{r \rightarrow 0^+} \sqrt{1+\cos \theta \sin^2 \theta}##. I'm not sure how to proceed

Wrong expression!

What is the definition of a limit of a two-variable function?

The limit of a two-variable function is the value that a function approaches as the input values get closer and closer to a particular point on a graph. It represents the behavior of the function at that point and can be used to determine the continuity of the function.

How is the limit of a two-variable function calculated?

The limit of a two-variable function is calculated by evaluating the function at specific points that approach the desired point on the graph. These points can be found by setting one variable to a constant and varying the other variable until it approaches the desired point. The resulting values are then used to determine the limit.

Can the limit of a two-variable function exist at a point where the function is not defined?

Yes, the limit of a two-variable function can exist at a point where the function is not defined. This is because the limit only considers the behavior of the function as the input values approach the point, not the actual value of the function at that point. However, if the limit does not exist, then the function is not continuous at that point.

What is the difference between a one-sided limit and a two-sided limit?

A one-sided limit of a two-variable function only considers the behavior of the function as the input values approach the desired point from one direction. This can be either from the left or right side of the graph. A two-sided limit, on the other hand, considers the behavior of the function as the input values approach the desired point from both directions.

How can the limit of a two-variable function be used in real-world applications?

The limit of a two-variable function can be used in a variety of real-world applications, such as determining the maximum or minimum value of a function at a particular point or finding the rate of change of a function at a specific point. It can also be used in physics, economics, and other fields to model real-world phenomena and make predictions based on the behavior of the function.

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