Multivariate Limit of (x^4 + y^4)/(x^2 + y^2) using Polar Coordinates

TL;DR

Solving a limit using polar coordinates, proving its existence regardless of direction, and finding the required limit.

Transcript

we're being asked to find the following limit so whenever you have to evaluate a limit the first thing you should always try is to see if plugging in the numbers works so if we plug in x equals zero and y equals zero we end up with zero plus zero divided by zero plus zero so zero over zero so it doesn't work but it's at least worth thinking about i... Read More

Key Insights

• 🥺 Plugging in numbers failed, leading to the necessity of polar coordinates for evaluation.
• ❣️ Utilizing the substitution of x and y with R and theta streamlined the limit-solving process.
• ⛔ The proof of limit existence from all directions ensures its accuracy and validity.
• 🔨 Applying the squeeze theorem as a validation tool showcases the stability and reliability of the limit evaluation.

Questions & Answers

Q: How is the use of polar coordinates beneficial in evaluating this limit?

Polar coordinates provide a substitution for x and y, allowing for a more structured approach to solving the limit and simplifying calculations.

Q: Why is it crucial for a multivariable limit to exist irrespective of the approach direction?

The existence of a multivariable limit from all directions ensures the validity of the limit's value, indicating consistency in its evaluation.

Q: How does the proof of independence of the limit from theta solidify the solution?

By demonstrating that the limit is constant regardless of the angle theta, the solution's reliability and accuracy are reinforced.

Q: Why is the squeeze theorem applied to validate the limit's existence in this scenario?

The squeeze theorem provides a logical framework to confirm the limit's convergence by comparing it to other bounded functions and establishing its stability.

Summary & Key Takeaways

• Attempted to find a limit by plugging in numbers, but resorted to polar coordinates.

• Utilized R squared as x squared plus y squared, leading to a substitution.

• Proved the existence of the multivariable limit regardless of the direction of approach.