L'Hopital's Rule and Infinite Limits

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SUMMARY

The discussion confirms that applying L'Hopital's Rule to limits resulting in an infinite value is valid, particularly in the context of limits in indeterminate forms such as 0/0 or infinity/infinity. The specific limit analyzed is \lim_{\substack{R\rightarrow 1}} \frac{RP'}{P}, where P is defined as P= c R J_1(\alpha R) - \frac{R^2 F}{\alpha^2}. The application of L'Hopital's Rule leads to the conclusion that \lim_{\substack{R\rightarrow 1}} \frac{RP'}{P}=\left[1+R\frac{P''}{P'}\right]_{R=1}\rightarrow \infty, validating the use of the rule in this scenario.

PREREQUISITES
  • Understanding of L'Hopital's Rule
  • Knowledge of limits and indeterminate forms
  • Familiarity with derivatives and their notation
  • Basic understanding of Bessel functions, specifically J_1
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  • Study advanced applications of L'Hopital's Rule in calculus
  • Explore the properties and applications of Bessel functions
  • Learn about the behavior of limits involving derivatives
  • Investigate other techniques for evaluating indeterminate forms
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Students and professionals in mathematics, particularly those studying calculus and differential equations, as well as anyone interested in advanced limit evaluation techniques.

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Say you have a limit in indeterminate form (0/0 or infinity/infinity) and you apply L'Hopital's rule to it and the result is an infinite limit. Is that a valid answer? Can L'Hopital's rule be applied in this way?
 
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Yes, that is valid.

Can you give us the limit to make sure we mean the same thing here??
 
Here's the limit I'm thinking of:

<br /> <br /> \lim_{\substack{R\rightarrow 1}} \frac{RP&#039;}{P},<br /> <br />

where primes are derivatives w.r.t. R. Also,

<br /> <br /> P= c R J_1(\alpha R) - \frac{R^2 F}{\alpha^2},<br /> <br />

where J_1 is a Bessel function of the first kind. Two of the three constants (c,alpha,F) are chosen such that P(1)=0 and P&#039;(1)=0 and the third is chosen for convenience. Thus the limit is in the form 0/0, so L'Hopital's rule leads to the following:

<br /> <br /> \lim_{\substack{R\rightarrow 1}} \frac{RP&#039;}{P}=\left[1+R\frac{P&#039;&#039;}{P&#039;}\right]_{R=1}\rightarrow \infty<br /> <br />
 
Ah yes. What you did is indeed a valid use of l'hospital's rule.
 

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