Limit of sqrt(x^2 + y^2)ln(sqrt(x^2 + y^2)) using Polar Coordinates and L'Hopital's Rule | Summary and Q&A

TL;DR

Using polar coordinates, the video demonstrates how to evaluate a limit that approaches zero by applying L'Hopital's rule.

Key Insights

• 🔌 It is essential to plug in the approaching values of variables when evaluating limits.
• 💁 A limit can sometimes result in an indeterminate form, where the value is not determined by direct substitution.
• 🐻‍❄️ Polar coordinates can be useful when simplifying and evaluating limits.
• 👻 L'Hopital's rule allows for the evaluation of limits involving indeterminate forms by taking derivatives.
• ⛔ The approach from which a variable approaches a particular value can impact the calculation of the limit.
• 🐻‍❄️ Rewriting the limit using polar coordinates and the natural logarithm facilitates the application of L'Hopital's rule.
• 🦖 The derivative of the natural logarithm of R is 1/R, and the derivative of 1/R^2 is -1/R^2.

Transcript

in this video we're going to use polar coordinates to evaluate a limit now when evaluating limits the first thing you should always do is plug in the values that your variables are approaching so here X is approaching zero and so is y and if we actually plug in 0 for both x and y we end up in a bit of trouble so we end up with zero times the natura... Read More

Questions & Answers

Q: How do you determine which values the variables are approaching when evaluating limits?

When evaluating limits, it is essential to identify the approaching values for the variables. In this case, both X and Y approach zero, leading to the use of polar coordinates to evaluate the limit.

Q: What is the significance of R approaching zero from the right?

As the pair XY approaches the origin (0, 0), the value of R, which represents the distance from the origin, also approaches zero. The direction from which R approaches zero, in this case from the right, is crucial for the calculation of the limit.

Q: Why is the use of polar coordinates recommended in this problem?

The decision to use polar coordinates is based on the observation that both X and Y can be expressed in terms of R in the polar coordinate system. This simplification allows for an easier evaluation of the limit.

Q: How does L'Hopital's rule come into play in evaluating the limit?

L'Hopital's rule states that if an indeterminate form is encountered, the limit can be evaluated by taking the derivative of the numerator and denominator. In this case, the natural logarithm of R is differentiated to facilitate the calculation of the limit.

Summary & Key Takeaways

• The video explains the process of evaluating limits by plugging in the approaching values of variables, highlighting the issue of an indeterminate form.

• By using polar coordinates, the limit is rewritten to incorporate R and the natural log of R, allowing the application of L'Hopital's rule.

• After taking the derivative and simplifying, the final limit is determined to be 0.