Solving an Indeterminant Limit Problem with L'Hopital's Rule

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SUMMARY

The forum discussion focuses on solving the limit problem using L'Hôpital's Rule. The limit in question is expressed as \(\lim_{x \to \infty} \frac{(8-x)^{200}}{64(3-x^2)^{100}}\). Participants agree that the expression is in an indeterminate form, suggesting the use of L'Hôpital's Rule, while emphasizing the importance of considering the highest order terms in both the numerator and denominator for simplification as \(x\) approaches infinity.

PREREQUISITES
  • Understanding of limits and indeterminate forms in calculus
  • Familiarity with L'Hôpital's Rule and its application
  • Knowledge of polynomial expansions and their significance in limit evaluation
  • Basic algebraic manipulation skills for simplifying expressions
NEXT STEPS
  • Study the application of L'Hôpital's Rule in various limit problems
  • Learn about polynomial expansion techniques for higher-order terms
  • Explore examples of indeterminate forms and their resolutions
  • Review advanced limit concepts, including asymptotic behavior of functions
USEFUL FOR

Students studying calculus, particularly those tackling limits and indeterminate forms, as well as educators looking for examples of L'Hôpital's Rule applications.

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Homework Statement



Find [tex]\lim_{x-> \inf} \frac{(8-x)^{200}}{8^{x+2}}*\frac{8^x}{(3-x^2)^{100}}[/tex]

Homework Equations


The Attempt at a Solution


Simplified to:

[tex]\lim_{x-> \inf} \frac{(8-x)^{200}}{64(3-x^2)^{100}}[/tex]

Indeterminant form, so I suppose L'hopital's, but it doesn't seem very efficient?
 
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Write the first two or 3 highest order terms of the expansions of the numerator and denominator. You'd agree that all lower order terms become increasingly insigificant as x becomes large, right?
 
Last edited:

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