Multivariable Function Limit by Squeeze Theorem

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SUMMARY

The discussion focuses on the application of the Squeeze Theorem to evaluate the limit of a multivariable function, specifically analyzing the expression r2cos4(θ) as r approaches zero. Participants confirm that the assumption that r2cos4(θ) approaches zero is valid due to the bounded nature of the cosine function, where |cos(θ)| ≤ 1. The conversation emphasizes the importance of simplifying inequalities before converting to polar coordinates to streamline the limit evaluation process.

PREREQUISITES
  • Understanding of the Squeeze Theorem in calculus
  • Familiarity with polar coordinates in multivariable calculus
  • Knowledge of trigonometric functions and their properties
  • Basic limit evaluation techniques in calculus
NEXT STEPS
  • Study the Squeeze Theorem applications in multivariable calculus
  • Learn about polar coordinate transformations in limit problems
  • Explore trigonometric function properties and their implications in limits
  • Practice limit evaluation techniques with various multivariable functions
USEFUL FOR

Students and educators in calculus, particularly those focusing on multivariable functions and limit evaluation techniques, as well as anyone seeking to deepen their understanding of the Squeeze Theorem.

Peacefulchaos
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I have uploaded my question along with all of my work to Google Docs as a PDF, it can be found https://docs.google.com/fileview?id...1ZDktZmVlYjA4MTQxNzE2&hl=en&authkey=CL-llawP", which is why I did not follow the template provided. (I already had it in a PDF :$)

I am curious if it is appropriate to assume that r^2*cos^4(theta) goes to zero when I am trying to find the minimum of the denominator. (You will see what I am talking about once you get to the step.) If you don't follow my work I'll be happy to explain my thought process.
 
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Your work is perfect, the assumption is valid since [tex]| \cos\theta | \le 1[/tex] and likewise for the sine function. You can save some work by working the inequality a little before switching polar.
 

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