# Evaluating a limit of a function of two variables

• Cristopher
In summary, the limit \displaystyle\lim_{(x,y)\to(-1,0)}\frac{y^4(x+1)}{|x+1|^3+2|y|^3} was proven to be 0 using Hölder's inequality. The conversation also mentioned the word DANG, which serves as a reminder of the four methods to approach limits: direct substitution, algebraic manipulation, numerical approximation, and graphing.
Cristopher
I want to evaluate $\displaystyle\lim_{(x,y)\to(-1,0)}\frac{y^4(x+1)}{|x+1|^3+2|y|^3}$

With some help, I was able to prove that the limit is 0, using Hölder's inequality. Like this:

$$\left(|x+1|^3\right)^{1/5}\left(\frac{1}{2}|y|^3\right)^{4/5}\leq\frac{1}{5}|x+1|^3+\frac{4}{5}\frac{1}{2}|y|^3$$

Raising to the $5/3$ power and cancelling we get:

$$|x+1|\left(\frac{1}{2}\right)^{4/3}y^4\leq\left(\frac{1}{5}\right)^{5/3}\left(|x+1|^3+2|y|^3\right)^{5/3}\\ \frac{|x+1|y^4}{|x+1|^3+2|y|^3}\leq\sqrt[3]{\frac{16}{3125}}\left(|x+1|^3+2|y|^3\right)^{2/3}$$

But I wonder if there are any other ways to prove it. Does anyone have other ideas?

Thanks for any input.

We used to use the word DANG to remind students of the four ways to approach limits:

Direct substitution
Algebraic manipulation
Numerical approximation
Graphing

## What is the definition of a limit of a function of two variables?

A limit of a function of two variables is the value that a function approaches as the two input variables approach a specific point. It represents the behavior of the function at that point and can be used to determine the continuity and differentiability of the function.

## How do you evaluate a limit of a function of two variables algebraically?

To evaluate a limit of a function of two variables algebraically, you can use the same techniques as for a single variable limit, such as direct substitution, factoring, and rationalizing. You may also need to use the properties of limits, such as the sum, difference, and product rules.

## What is the significance of evaluating a limit of a function of two variables?

Evaluating a limit of a function of two variables allows us to understand the behavior of the function at a specific point and make predictions about its behavior in the surrounding points. It also helps us determine the continuity and differentiability of the function, which are important concepts in calculus.

## Can a limit of a function of two variables exist even if the function is not defined at the point?

Yes, a limit of a function of two variables can exist even if the function is not defined at the point. This means that the function is approaching a specific value as the two input variables approach the point, but the function may not be defined at that point itself.

## What are some common techniques for evaluating limits of functions of two variables graphically?

Some common techniques for evaluating limits of functions of two variables graphically include creating a table of values, plotting the function on a graph and visually estimating the limit, and using the concept of a limit to find the behavior of the function as the two input variables approach the point on the graph.

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